I have both a slim and a 360 so my recommendations include multiplat games I didn't actually play on ps3 but still should be awesome (no shooters included);
PS3 exclusives
1) Uncharted + Uncharted 2 (great platforming, combat and just generally heaps of over the top absurd fun. Top notch graphics too)
2) God of war collection + God of war 3 (again top platforming/action, super gory and 2 of the best ps2 games remastered in 1080p. Some puzzles are very challenging though, don't know if that is your thing)
3) Heavy Rain (May not initially catch your eye but I really thoroughly enjoyed this game, the storyline and gameplay are really immersive and it's overall a really unique experience)
Multiplats
1) Batman Arkham Asylum (I absolutely love this game, even if you aren't particularly into batman I would definitely recommend it. Fantastic combat, exploration, puzzles, gadgets, collectibles and replayability. Plus the voice acting especially Mark Hammil as the joker is just perfection. Best game of 09 imo)
2) GTA IV + dlc (Although it has been out a few years still absolutely great fun and the new dlc especially ballad of gay tony means you get 3 games for around the price of 1)
3) Deadspace (Great creepy atmosphere, enjoyable gameplay and really unique story/background. The tension is intense in this game)
4) Assassins creed 2 (Just plain old medieval fun and unlike the first one it isn't incredibly repetitive and doesn't become tedious after a while. I'd definitely recommend at least reading the storyline of number 1 first though as otherwise you might be slightly confused with what is happening in assassins creed 2)
5) Resident Evil 5 (Only buy this game if you have someone to play it with, 1 player sucks but co-op is the ****. Maybe slightly dated gameplay but is really the best co-op game I've played this gen. Highly underrated)
6) Not played it yet but have red dead redemption being posted to me, I've heard nothing but good things about it so will say check it out.
Games I'd maybe avoid that others will recommend
1) Metal Gear Solid 4 (Woooo controversial but to be honest I didn't like it. If you are a newbie to the metal gear universe like I was then this game isn't kind, it doesn't explain anything and you pretty much sit there with no idea of what is going on for most the game. I prefer simplicity in controls and gameplay in any game I'm playing and this has neither, to be honest I found it overly complicated and boring)
2) Killzone 2: (I know you said no shooters but since this is a ps3 exclusive I though you may be tempted to buy it. Yes it has nice graphics but compared to other shooters the gameplay is sluggish and frustrating, and the controls suck. It does have its moments and is quite fun, but really does pale compared to other shooters. Lightning gun though is brilliant)
3) Infamous (Beaten the game but still not sure whether I liked it or I didn't. Yes the powers are fun and some of the storyline/missions are good but to be honest it felt like a half finished game. There were glitches galore and it was quite repetitive and tedious in parts. Has a decent amount of gametime but I reckon there are a lot better sandbox games (basically the rockstar ones) and superhero ones (batman))