Might be able to get away with RTS games, there is violence but you see very little of it being high above the combat. Will take some convincing though.
Sins of a Solar Empire is an easy option if your into strategy.
There are city builder games like Sim City 4 (of you want to brave the server situation) or Tropico 4 (Think a Cuban dictator simulator
Aforementioned Valve games and Portal series, they really are very good, no violence games that looks very nice.
Definite +1 to Indie games, they wont push the graphics card in any way but are very good and often have great aesthetics. I suggest looking into Braid, LIMBO, Machinarium, Super Meat Boy and the BIT.TRIP series. You might even be able to buy Humble Bundles off people, they have a ton of good games. Heard Monaco: Whats Yours is Mine and Bastion are fairly good as well.
FEZ is coming to the PC soon, look into that as well.
(So many indie games now!).
EDIT: Yet another. Have look at a game called Antichamber, very Portal-esque first person puzzler that defies everything you know about physics and logic.
On that same thought, Quantic Conundrum is another very similar game.
Racing games I have little experience in, but I did enjoy Burnout: Paradise City if it helps.
Minecraft and Terraria for your games to sink endless hours into creating something.
You could try games like Torchlight II, very cartoony aesthetic and plays like Diablo III.
Dont forget you can always download emulators and play the older console games. Ocarina of Time and Majora's Mask might just have to serve for your action adventures, very few nowadays that arent violent in some way.