If you don't have mining or want to spend time mining then I have to recommend buying these things off the Auction House.
Since patch 2.4 was implemented you can now train up to Blacksmithing at any trainer available. To get your skill past 300 you will need to see the Master Blacksmithing Trainer that is located in each factions appropriate starting city for the Burning Crusade in Hellfire Peninsula, Honor Hold and Thrallmar.
This guide is the quickest way to get your Blacksmithing to 375, it does not contain information about specializing between Weaponsmithing and Armorsmithing, but will provide steps (when necessary) for each specialization in order to level.
1-75 only requires you to produce three different items and they are produced with the simplest of components. Keep your Rough Grinding Stones because they will be used later in this guide. The Rough Sharpening Stones can be used at alts and you can vendor trash the Copper Chain Belts.
This step is just a little bit more complex than the previous. There is a total of 5 items that you will be crafting and only one of them requires 2 components to craft (I hope you saved your Rough Grinding Stones from the previous step). This time ensure that you save all of the Heavy and Coarse Grinding stones you make because they are used later in the guide as well. This section will take you from 75-150. The Silver Rod you will be making can be sold on the Auction House to enchanters.
The guide becomes a little bit more difficult here, you will have a lot of 2 and 3 component crafting combines, but it's not too bad. This step will get you from 150-215. Green Dye can be bought at any tailoring or most of the general goods vendors. If you kept your grinding stones from the previous steps than you will have a head start on getting the materials required to level in these section. Make sure you save your Solid Grinding Stones.
This part of the guide will require you to make 7 different items to get from 215-300. The first item requires your Solid Grinding stones that you should have saved from previously. There are some components that may be expensive, like Thorium Bar's and Core of Earth, but this is the fastest way to do this. Also, you will get a choice from 260-270, depending on whether or not you are a Weaponsmith or Armorsmith. The Earthforged Leggings and Light Earthforged Blade can be learned from the respectable Weaponsmith or Armorsmith trainer, usually located in one of the major cities.
Now you are ready to start on the new Burning Crusade recipes. This step will take you from 300-360 and then it is preferred that you use either Aldor or Scryer to go from 360-375. To go from 300-360 you will be making 8 different items. Some of them may sell well on the Auction House, like, Lesser Rune of Warding and Lesser Ward of Shielding for Tanking. You can try selling the other items as well, chances are there are a market for these.
It has been reported that follow patterns are limited supply from the following vendors, we are unsure if this is true or not so if you can't find the pattern at these merchants when you go there, then you may have to check back periodically.
You can get the Adamantite Cleaver recipe from the Lower City Blacksmithing Supply Merchant in Shattrath, his name is Aaron Hollman.
The Lesser Ward of Shielding comes from Rohok in Thrallmar for Horde. For Alliance, Mari Stonehand in Wildhammer Stronghold has the recipe.
Adamantite Weightstone comes from Zangarmarsh at the Cenarion Expedition camp. The merchant's name is Fedryen Swiftspear and you must be honored with the Cenarion Expedition to obtain this recipe.
There are two different patterns you will be required to get to go from 360-375 as Aldor and the components required to make each one are not cheap. Primal Water and Primal fire can go anywhere from 5-30g each, depending on your server and you will need quite a bit for each try, but this is the best way to go. The Flamebane gloves can be purchased from the Aldor Quartermaster when you are honored and the Flamebane Breastplate will require you to be revered with Aldor.
There is only one item that will be made to level up from 360-375 as Scryer. The Enchanted Adamantite Belt only requires you to be friendly, however, the materials are going to be pretty expensive per attempt. Arcane Dust and Large Prismatic Shards do not usually run cheap so I hope you know or have an enchanter.