I am looking for some quirky, outside the box type games to play. I always like playing games like the Katamari Damacy, Pixel Junk Monsters and an old PS1 game No One Can Stop Mr. Domino. Does anyone know any other games like these that are not your typical FPS and Action games?
I could probably think of a few oddball games out there. One that comes to mind is Datura on the PlayStation Network. The game has been on sale recently for a couple of bucks. The sale might be still going on for at least a couple of more days. Specifically, in this game you control a floating hand as you make your way through a forest and come across different people and puzzles you can interact with along the way. I haven't played the game myself, but it's said to be a strange one.
Another peculiar fix might be that of Quantum Conundrum. Much like the Portal series (which itself is somewhat unusual), this game gives you a scientific weapon of sorts to complete a series of conundrums involving weight. You can make the cubes light as a fluffy pillow or as heavy as a barrel full of bricks.
There are various other games that may appear to you as out of sorts, but then again maybe they're too "normal" for your tastes. Other games I think of off the top of my head consist of everything from Papo and Yo (a platformer where you must guide a rhinoceros to the end goal), From Dust (a God game where you shape the environment to get the people to safety), and The Unfinished Swan (another platform/puzzling game where you'll use your paintbrush powers to make your way through the levels).
Just to repeat what tross88 said really - the PSN is the place to look for quirky or innovative games. Limbo and Braid are some more examples of games with both original ideas and great gameplay.
As far as full price games go, have you played the recent Xcom game? It's an excellent turn based strategy game and there arn't many of those being made these days! Ther's also nothing else quite like the puzzle game Portal 2 and it's one of the best games this gen.