As others said, it depends upon your role... DM vs a spellcasting PC vs a martial PC. Also, the higher the level of the game, the more dice you need... casters get 1 dice of damage per caster level, in many cases, and clerics heal more and more as they level up. You might just order a "pound of random dice" and you'll probably get enough of what you need.
So, with that in mind, let me go over the dice:
d4: used for daggers, magic missiles, and burning hands, among other things. If you are a non-caster, 2d4 is enough. Arcane casters should have 5d4.
d6: very common die, used for many weapons, many spells, and sneak attack damage, healing bursts, etc. The more you have, the better. Non-casters can get away with 5d6, casters should have 10d6 or more.
d8: common for many weapons, curing spells, and some attack spells. Non casters can use 2d8, casters should have at least 4d8 or more.
d10: used for larger weapons, not very common in spells. 2d10 should be fine.
d12: not a commonly used die except for some heavy weapons. 1d12 should be ok.
d20: very commonly used dice, of course... the system is named after it. But you don't often need to roll a lot at once. 2d20 should be fine for most people.