I'll start this thread with some basic tips based on my little personal experience ( EDIT: cleared the game on hc with a Shaman!), everybody's contribution and experience are really welcome!!
First protip:
DON'T DIE!! 
But you know that already I'm sure, so the point is: how? Well here's a few suggestions...
- don't be too cocky, so don't start a fight with your big/emergency skills/spells on CD, don't pull more/longer than you can handle, remember that a double ambush ( in Nightmare/Hell)is always a possibility, slow down if you don't know what mobs to expect in the zone ( in this case could be wise to use a sc toons to learn more).
- gear defensively, look for stats like +all res, -status reduction (-max stun is a must in hc, -fear close second, -silence is mandatory for casters, -chill is good but can be skipped), life/mana leech (as per 1.0.75 all kind of damage and even DoTs can leech), then look for +all attr ( +skill is meh), +health/mana; all of that without sacrificing too much dps/haste, because the best defence is- also- a good offense!
- prepare a boss fight beforehand, gear accordingly to what to expect (what skills/spells it uses, how many adds will pop up etc...) and have a strategy.
- always look out for some nasty affixes and be cautious when a champ pops up - e.g. I've died twice with high lvl Wizards due to vicious champs casting Iron Maiden right after I cast Charged Bolts
- learn from your own mistakes! I now know to freeze mobs that can cast Iron Maiden before casting Charged Bolts, for example.
- grind for gear if you feel the need to (when your killing speed is getting low or you're getting squishy); this is true mostly when you first start Hell and have few/none elite pieces, you can grind for them in Nightmare Nimgaul.
- spend your gold upgrading the gear you're using, even if only for a few upgrades, that +arm +allres or +dmg could make the difference between life and death.
- look for help in the wiki/forums -I did- lots of great info and tips out there.
That's all I can think of now, I'm missing something for sure, so please add to this with your own tips/experiences.
