"Notched Pickaxe" by Marco Hazard is licensed under CC BY-SA 2.0
This isn't too exciting, but it's necessary. You can find coal on the side/tops of mountains, in caves, and if you dig down a few blocks you'll probably find some.
Iron is a bit more challenging to find than coal, but it's pretty easy. You can find it in caves not that far below ground, by digging down like 15 blocks, and very rarely in mountains.
Gold is near useless. Basically the only use is for decoration and golden apples. Gold can be found below Y level 30, and is pretty rare. Redstone and Lapis spawn lower, but are more common. The best way to find them is in caves and while "strip mining"
Diamonds can be found best while "strip mining" at Y level 11. Same with emeralds, but emeralds are extroardinaly rare. They are also near useless. Emeralds can also be obtained by trading with villagers.