Engineering supplies mainly come from mining, if you don't have a miner then you will have to buy them. The fastest way to level up your engineering is to buy these components off the Auction House when available.
Since patch 2.4 was implemented you can now train up to Engineering at any trainer available. To get your skill past 300 you will need to see the Master Engineering 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 Engineering to 375. Most people choose this profession for PvP trinkets and the Gnomish Flying Mount, but there is an array of other items to choose from. This guide does not cover any of the engineering specializations and no specialization is required to complete this guide. Just for your information, the specializations are Gnomish or Goblin.
There are going to be a lot of things you craft that will be required to craft other things. Make sure you keep your Copper Tube's and Handful of Coppert Bolts from this first section because they are both used in creating the Rough Broomstick.
From 75-150 you will be making things that play an important role as components for other things. Silver contact will not only be used in this section but also the next. You will probably not make enough while skilling up on them and have to buy some off the auction house or make more later on, either way, keep the ones you make. It is also a good idea you holdon to the Blasting Powders you make becuase those are common components in other recipes. Remember anything you end up buying from the Auction House you can also make and sell on the Auction house.
The very first item you make in this part is the Red Firework's. You can have some fun and launch a few of these babies. If you haven't tried out any of the Bombs you've been making you can have fun with these on lower level mobs. And if you are a Hunter or someone who uses bullets, engineering will come in handy to craft them as you see in this section. Save your Blasting Powder.
You start the 225-300 section off by making Mithril Casing. This item used to be in semi-high demand because it was used for a quest in Ungaro Crater, the demand is probably down but I bet you could sell some of these. You will not need to keep your Dense Blasting or any of the previous powders any longer. The next section is Burning Crusade specific and almost all the components are made from Burning Crusade engineering recipes.
The first part from 300-310 is a multiple choice. You can do whatever recipe you want to 310 but we want to point out that all of these will be used as components for some of the other recipes. So if you don't want to buy them off the auciton house prepare to make multiples of all of these.
Once you reach 360 we recommend you find the Khorium Scope Schematic. It is a rare, bind on pickup recipe so you must have Engineering as a profession in order to see it drop from the mob. The schematic drops from Sunfury Bowman in Netherstorm and has a drop rate of about 1 in 500, so be prepared to be out there killing for awhile. The Sunfury Bowman are located around Manaforge Duro.