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.
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, from common to more rare: copper, tin, silver, iron, gold, mithril, and more. To find rare veins, you will have to fight your way (or sneak) to more powerful regions of the world. The first veins you will come across are copper. As you head to more powerful areas, you will begin to see tin and so on.

Using Mineral Veins

After you find a mineral vein, right-click on it to begin mining. If you are successful, you will now be able to loot some minerals and possibly some rough stones. If you fail, try again. Continue to mine the mineral vein until it disappears.

Mining Components

When you mine, you will find three main types of ingredients: metal, stones, and gems. Gems are rare and are used for better recipes for other professions, such as blacksmithing or engineering. Collect all the ingredients you find and keep them for later. Stones can be transformed without the use of a forge. Many of these components can also be found from killing monsters or bought from other players.
When you mouse over the mineral vein, you will see a color showing whether you will gain skills by mining it.

Gray - You cannot increase your skill by mining.
Green - These mineral veins give you a fairly poor chance to increase your skill. You can still do it, but its going to take a lot of gathering/creation to up your skill with these mineral veins.
Yellow - These mineral veins give you a reduced chance of increasing your skill. Its still pretty reasonable to try to increase your skills with these mineral veins.
Orange - These mineral veins give you your best chance of increasing your skill. For many of the Professions making an orange item will always increase your skill.
Red - you cannot mine it yet. You need a higher mining skill.
After you have mastered the mining of a particular type of mineral, you will no longer gain skills by mining it. However, you can still use the minerals and stones you mine.

Using a Forge/Smelting

Now that you have the precious metal, you can smelt it into bars that can be used for creating armor, weapons, or other items through the blacksmithing or engineering skills.
Find yourself a forge. They are located in main towns and some smaller towns. If you cant find one, ask around to determine their locations. To confirm you have in fact found a forge, move your mouse over the location and see if it says "forge". You can also try smelting the metals. You do not need to interact with the forge; you only need to be near it.

Now that you have found a forge, its time to smelt the metal. Open up your abilities book and locate the smelting icon. Press that to see your recipes. Some examples of what you can create with smelting are: converting copper ore into copper bars and creating bronze by combining copper bars and tin bars. To gain additional smelting recipes/options, you will need to learn them from a mining trainer. The mining screen will tell you how many of each bar that you can smelt based on the types and amounts of raw material that you have.

Claim Jumping

You will undoubtedly run into another player who is after the same mineral node as you; in fact you might not even see each other before you are both mining it. This is a place where you have the opportunity to be kind or greedy or even-handed with your character. You might offer the other person the mineral node, or try to take it all for yourself, or even offer to alternate mining at the node to increase both of your skills.