This is what to bring when going mining so that you will be well prepared:
1. Pickaxe (or 2, depending on the durability)
2. Torches (at least one stack)
3. Sword (also bow and arrows is useful)
4. Bucket of water (in case for lava)
5. Extra blocks (dirt or cobblestone)
6. Shovel
7. Enderchest (not until 1.3)
8. Chest (to store items temporarily)
9. Flint n' Steel (let mobs burn)

This is also great because it fills all of your hotbar slots