So you want to be a miner? Mining is an exciting Profession used to gather materials for blacksmithing or engineering. However, you dont need those Professions to be a miner. You could simply collect resources and sell them to blacksmiths and engineers. You're going to find these spots anyway as you travel throughout the world, so why not cash in on them? However, most people combine mining and blacksmithing or mining and engineering.

Becoming a Miner

First you will need to find the mining trainer. After you have learned basic mining, open up your abilities book (hit "p") and look for the "Find Minerals" icon that has been added to your abilities. This will be used to detect mineral veins. Shift Click and drag that icon over to one of the rows of your action bar.
Now you need to go buy yourself a "mining pick". They are usually sold by a "trade merchant" or other merchants near the mining trainer.

Finding Mineral Veins

Now you need to find mineral veins. While you are playing, hit your "Find Minerals" Icon (or press the corresponding hot key). Mineral spots will now show up on the mini-map when you are near them. Seek them out. You will have to look around to find mineral veins. They are often found near quest locations and inside mines. They are usually near mountain terrain. It's up to you to discover the good mining locations. Often it's good to find a path that you can repeat between the mining nodes you have discovered. If a mining node is gray on the map, that means it's above or below you or possibly in a cave nearby.
Have "Find Minerals" active nearly all the time so that you will know when minerals are nearby.
Mineral Vein Types

Minerals come in many different versions. The first veins you will come across are Copper. To find more valuable and higher level ore, you will need to travel to higher level zones. For example Elwynn Forest/Durator has Copper. To find Tin you will need to travel to Westfall/The Barrens. Look below for a list of locations.