Horde Armorsmithing
As an FYI every smith can complete 90% of this quest chain without dedicating themselves to armorsmithing. In fact I highly recommend it mostly for the 2 side rewards you get from doing it (see the notes near the end).

Armorsmithing is not for the faint of heart or those unsure if it's the direction they really want to go. It is a long drawn out process that will take a significant amount of time, effort, and likely money to complete. But in the end it is sooooooo worth it for those who can use the gear or who have designs on becoming their guild's Armorsmith. After reading this guide you'll likely understand why armorsmiths charge so much for their skills.

For those who are determined to become armorsmiths I have a few recommendations.
1. Begin this process as soon as you hit 40
2. Hold on to every scrap of iron and mithril you get your hands on. You will need every bar. Also hang on to as much heavy and solid stone (ironically obtained from mining iron and mithril) as you can as well. And keep any gems you happen to get since several of the quests and recipes require them.
3. For those who choose to get their metals the old fashioned way and collect it themselves, I recommend Arathi Highlands for you iron needs. Simply make a circuit of the outer rim moving north from Hammerfall. I personally would queue up for AB at the Defilers camp behind Hammerfall and grind iron between bgs. The iron was usually respawned by the time I got out. (Also a useful tip for those needing to collect iron for Neutral to Friendly turn-ins for Thorium Brotherhood). Mithril and (to a lesser extend iron) can be easily be farmed by doing a similar circuit of the Badlands moving southeast from Kargath. Desolace is an alternate to the Badlands but it seems to have a wider variety of minerals I would recommend sticking to the Badlands for mining consistancy.
4. For those who want to take the easy route. Have A LOT of gold you're not attached to ready and a good fast way to make a lot more. Iron an mithril are some of the more expensive of the easily farmed metals. This is mostly because they are used in a wide variety of professions and (especially in the case of iron) faction rep turnins.
5. Once you complete one part of this guide, if the quest or person mentioned next does not offer any quests, make sure your blacksmithing is of the appropriate level for the crafted items needed and/or you may not be high enough level.. Skill up your smithing 5 points and or go level up once or twice and check back with the NPC mentioned again.

If you're still interested in becoming and armorsmith, then lets begin...

Once your smithing is 215+ head to Booty Bay and pick up recipes for Steel Breastplate (200 skill) and Steel Plate Helm (215 skill) from Brikk Keencraft in the Master Blacksmith in Booty Bay. He can be found by heading straight north from the dock up the ramp. You will see the smithy as you continue to walk north past the exit leading out of Booty bay into Stranglethorn. Once you have these two recipes go to the Blacksmith area in Orgrimmar (found in the Valley of Strength past the Battlegrounds hall).
Aturk the Anvil should have the following quest available:
1 - The Old Ways
Items needed: 4 Steel Breastplate, 4 Steel Plate Helm (both learned from Breek Keencraft in the booty bay smith)
Mats needed: 120 Steel Bars (120 Iron Bars + 120 Coal), 3 heavy grinding stone (9 heavy stone), 4 solid grinding stone(16 solid stone)

Ox will then whisper asking you to speak with him. (He's the Mithril Order Tauren standing next to Aturk). He then gives you the quest Booty Bay or Bust! Sending you to Booty Bay to speak with McGavan, who can be found standing outside the Booty Bay smith (where you learned the Steel Breastplate and Steel Plate Helm recipes).
McGavan will then send you to find Galvan the Ancient. Galvan will give you 3 quests whose rewards are plans needed in the next portion. Depending on your level and/or smithing skill, 1 to all 3 of these quests may be available at the same time. Seeing as you will need to finish all 3 I recommend obtaining all of the mats for all 3 before heading off to find Galvan, it will save you a lot of time running/riding/flying back and forth to the middle of Stranglethorn Vale to a town with an auction house/grinding ore.
1- Smelt On, Smelt Off
Items needed: 40 iron bars, 40 mithril bars
Rewards: Plans for Ornate Mithril Pants
2 - The Great Silver Deceiver
Items needed: 40 mithril bars, 5 truesilver bars
Rewards: Plans for Ornate Mithril Gloves
3 - The Art of the Imbue
Items needed: 40 mithril bars, 4 citrine
Rewards: Plans for Ornate Mithril Shoulders
Total mats needed for this portion: 120 mithril bars, 40 iron bars, 5 truesilver bars, 4 citrine

* * * E X T R A * * *
I foolishly started this at 55 and took the fast route to 215 smithing by grinding out only the cheapest skills that were orange to get my skill up and skipping all other blacksmithing quests so it may be different at lower levels.
If you have completed the other horde smithing quests you will be offered the quest Expert Blacksmith! for which your reward is the Signet of Expertise. This trinket has a 4 hour cooldown (3 for all intents and purposes) and creates the Hammer of Expertise a level 40 1-hander with the damage range and dps of a level 50 blue axe that lasts for 1 hour. While the hammer can't be enchanted you can slap a weightstone on it :)
* * * E X T R A * * *

Once completed Galvan will give the quest Galvan's Finest Pupil which sends you to find Trenton Lighthammer in Gadgetzan. Trenton then begins a new chain of quests. At this point, before you head to Gadget, make sure your smithing is at 235+ then stop in Booty bay to pick up recipes for Mithril Coif (230 skill), Heavy Mithril Boots (235 skill), and Heavy Mithril Breastplate (230 skill)
1 - A Good Head On Your Shoulders
Items needed: 2 Mithril Coif, 1 Ornate Mithril shoulder
Mats needed: 32 mithril bars, 12 mageweave cloth, 6 thick leather
Rewards: Ability to craft Ornate Mithril Helm
2 - The World At Your Feet
Items needed: 2 Heavy Mithril Boots, 1 Ornate Mithril Pants
Mats needed: 40 mithril bars, 1 truesilver bar, 1 solid grinding stone (4 solid stone), 1 aquamarine
Rewards: Ability to craft Ornate Mithril Boots
3 - The Mithril Kid
Items needed: 2 Heavy Mithril Breastplate, 1 Ornate Mithril Gloves
Mats needed: 46 mithril bars, 6 mageweave cloth, 1 truesilver bar, 1 solid grinding stone (4 solid stone)
Rewards: Ability to craft Ornate Mithril Breastplate
Total mats needed for this portion: 118 mithril bars, 18 mageweave cloth, 2 truesilver bar, 2 solid grinding stone (8 solid stone), 1 aquamarine

Once those are complete you now have the recipes needed to become an armorsmith

* * * E X T R A 2 * * *
Talk to Trenton again and he will give the quest Did You Loose This? (which is available to both sides BTW).
Just accept and talk to him again and he yells about your awesomeness as a smith :) and gives you the Glimmering Mithril Insignia (which is an awesome trinket for PVP).
* * * E X T R A 2 * * *

Now that you have all of the ornate mithril recipes, return to the smith area in Orgrimmar and speak to Okothos Ironrager who should now offer the following quest
1 - The Art of the Armorsmith
Items needed: 4 Ornate Mithril Helm, 2 Ornate Mithril Boots, 1 Ornate Mithril Breastplate
Mats needed: 108 mithril bars, 18 truesilver bars, 8 thick leather, 7 solid grinding stone (28 solid stone), 4 black pearl (most commonly found in Big-Mouth Clams), 2 aquamarine, 1 heart of fire

Gratz, you are now an armorsmith and should also have 2 very nice/useful trinkets

Total Mats needed(including crafting items): 346 mithril bar, 160 iron bar, 120 Coal, 25 truesilver bar, 18 mageweave cloth, 13 solid grinding stone (52 solid stone), 8 thick leather, 4 citrine, 4 black pearl, 3 heavy grinding stone (9 heavy stone), 3 aquamarine, 1 heart of fire.

Horde Armorsmithing
As an FYI every smith can complete 90% of this quest chain without dedicating themselves to armorsmithing. In fact I highly recommend it mostly for the 2 side rewards you get from doing it (see the notes near the end).

Armorsmithing is not for the faint of heart or those unsure if it's the direction they really want to go. It is a long drawn out process that will take a significant amount of time, effort, and likely money to complete. But in the end it is sooooooo worth it for those who can use the gear or who have designs on becoming their guild's Armorsmith. After reading this guide you'll likely understand why armorsmiths charge so much for their skills.

For those who are determined to become armorsmiths I have a few recommendations.
1. Begin this process as soon as you hit 40
2. Hold on to every scrap of iron and mithril you get your hands on. You will need every bar. Also hang on to as much heavy and solid stone (ironically obtained from mining iron and mithril) as you can as well. And keep any gems you happen to get since several of the quests and recipes require them.
3. For those who choose to get their metals the old fashioned way and collect it themselves, I recommend Arathi Highlands for you iron needs. Simply make a circuit of the outer rim moving north from Hammerfall. I personally would queue up for AB at the Defilers camp behind Hammerfall and grind iron between bgs. The iron was usually respawned by the time I got out. (Also a useful tip for those needing to collect iron for Neutral to Friendly turn-ins for Thorium Brotherhood). Mithril and (to a lesser extend iron) can be easily be farmed by doing a similar circuit of the Badlands moving southeast from Kargath. Desolace is an alternate to the Badlands but it seems to have a wider variety of minerals I would recommend sticking to the Badlands for mining consistancy.
4. For those who want to take the easy route. Have A LOT of gold you're not attached to ready and a good fast way to make a lot more. Iron an mithril are some of the more expensive of the easily farmed metals. This is mostly because they are used in a wide variety of professions and (especially in the case of iron) faction rep turnins.
5. Once you complete one part of this guide, if the quest or person mentioned next does not offer any quests, make sure your blacksmithing is of the appropriate level for the crafted items needed and/or you may not be high enough level.. Skill up your smithing 5 points and or go level up once or twice and check back with the NPC mentioned again.

If you're still interested in becoming and armorsmith, then lets begin...

Once your smithing is 215+ head to Booty Bay and pick up recipes for Steel Breastplate (200 skill) and Steel Plate Helm (215 skill) from Brikk Keencraft in the Master Blacksmith in Booty Bay. He can be found by heading straight north from the dock up the ramp. You will see the smithy as you continue to walk north past the exit leading out of Booty bay into Stranglethorn. Once you have these two recipes go to the Blacksmith area in Orgrimmar (found in the Valley of Strength past the Battlegrounds hall).
Aturk the Anvil should have the following quest available:
1 - The Old Ways
Items needed: 4 Steel Breastplate, 4 Steel Plate Helm (both learned from Breek Keencraft in the booty bay smith)
Mats needed: 120 Steel Bars (120 Iron Bars + 120 Coal), 3 heavy grinding stone (9 heavy stone), 4 solid grinding stone(16 solid stone)

Ox will then whisper asking you to speak with him. (He's the Mithril Order Tauren standing next to Aturk). He then gives you the quest Booty Bay or Bust! Sending you to Booty Bay to speak with McGavan, who can be found standing outside the Booty Bay smith (where you learned the Steel Breastplate and Steel Plate Helm recipes).
McGavan will then send you to find Galvan the Ancient. Galvan will give you 3 quests whose rewards are plans needed in the next portion. Depending on your level and/or smithing skill, 1 to all 3 of these quests may be available at the same time. Seeing as you will need to finish all 3 I recommend obtaining all of the mats for all 3 before heading off to find Galvan, it will save you a lot of time running/riding/flying back and forth to the middle of Stranglethorn Vale to a town with an auction house/grinding ore.
1- Smelt On, Smelt Off
Items needed: 40 iron bars, 40 mithril bars
Rewards: Plans for Ornate Mithril Pants
2 - The Great Silver Deceiver
Items needed: 40 mithril bars, 5 truesilver bars
Rewards: Plans for Ornate Mithril Gloves
3 - The Art of the Imbue
Items needed: 40 mithril bars, 4 citrine
Rewards: Plans for Ornate Mithril Shoulders
Total mats needed for this portion: 120 mithril bars, 40 iron bars, 5 truesilver bars, 4 citrine

* * * E X T R A * * *
I foolishly started this at 55 and took the fast route to 215 smithing by grinding out only the cheapest skills that were orange to get my skill up and skipping all other blacksmithing quests so it may be different at lower levels.
If you have completed the other horde smithing quests you will be offered the quest Expert Blacksmith! for which your reward is the Signet of Expertise. This trinket has a 4 hour cooldown (3 for all intents and purposes) and creates the Hammer of Expertise a level 40 1-hander with the damage range and dps of a level 50 blue axe that lasts for 1 hour. While the hammer can't be enchanted you can slap a weightstone on it :)
* * * E X T R A * * *

Once completed Galvan will give the quest Galvan's Finest Pupil which sends you to find Trenton Lighthammer in Gadgetzan. Trenton then begins a new chain of quests. At this point, before you head to Gadget, make sure your smithing is at 235+ then stop in Booty bay to pick up recipes for Mithril Coif (230 skill), Heavy Mithril Boots (235 skill), and Heavy Mithril Breastplate (230 skill)
1 - A Good Head On Your Shoulders
Items needed: 2 Mithril Coif, 1 Ornate Mithril shoulder
Mats needed: 32 mithril bars, 12 mageweave cloth, 6 thick leather
Rewards: Ability to craft Ornate Mithril Helm
2 - The World At Your Feet
Items needed: 2 Heavy Mithril Boots, 1 Ornate Mithril Pants
Mats needed: 40 mithril bars, 1 truesilver bar, 1 solid grinding stone (4 solid stone), 1 aquamarine
Rewards: Ability to craft Ornate Mithril Boots
3 - The Mithril Kid
Items needed: 2 Heavy Mithril Breastplate, 1 Ornate Mithril Gloves
Mats needed: 46 mithril bars, 6 mageweave cloth, 1 truesilver bar, 1 solid grinding stone (4 solid stone)
Rewards: Ability to craft Ornate Mithril Breastplate
Total mats needed for this portion: 118 mithril bars, 18 mageweave cloth, 2 truesilver bar, 2 solid grinding stone (8 solid stone), 1 aquamarine

Once those are complete you now have the recipes needed to become an armorsmith

* * * E X T R A 2 * * *
Talk to Trenton again and he will give the quest Did You Loose This? (which is available to both sides BTW).
Just accept and talk to him again and he yells about your awesomeness as a smith :) and gives you the Glimmering Mithril Insignia (which is an awesome trinket for PVP).
* * * E X T R A 2 * * *

Now that you have all of the ornate mithril recipes, return to the smith area in Orgrimmar and speak to Okothos Ironrager who should now offer the following quest
1 - The Art of the Armorsmith
Items needed: 4 Ornate Mithril Helm, 2 Ornate Mithril Boots, 1 Ornate Mithril Breastplate
Mats needed: 108 mithril bars, 18 truesilver bars, 8 thick leather, 7 solid grinding stone (28 solid stone), 4 black pearl (most commonly found in Big-Mouth Clams), 2 aquamarine, 1 heart of fire

Gratz, you are now an armorsmith and should also have 2 very nice/useful trinkets

Total Mats needed(including crafting items): 346 mithril bar, 160 iron bar, 120 Coal, 25 truesilver bar, 18 mageweave cloth, 13 solid grinding stone (52 solid stone), 8 thick leather, 4 citrine, 4 black pearl, 3 heavy grinding stone (9 heavy stone), 3 aquamarine, 1 heart of fire.

Allot of work but better to have it all ready
instead of flying back and forth for mats...
Here is the total of what you need of all mats to
become armorsmith form the start to finish...

Mithril Bars: 342
Truesilver bars: 23
Solid Grinding stones: 9
Thick leather: 26
Aquamarine: 2
Heart of fire: 1
Iron bars: 40
Mageweave cloth: 18
Citrine: 4

Now, go grind what you need and you'll become
an armorsmith even faster!

Note: Try to have some extra of all mats just in case ;)