Well, you can download PSOne and PS2 Classics from the Playstation Store, but if you're looking to play your own old games, you're fresh out of luck. The PS3 is backwards compatible with most PSOne games, but for PS2, you'd need an older fat model PS3, and even then it gets interesting. Depending on the year, model and region, the PS2 compatibility can be hardware or software based, both of which vary in compatibility rates.
I know for a fact that the European 60 Gb fat model is PS2 compatible, but how I do not know.
So all in all, on a slim PS3, you'd best hit the Playstation Store or start digging your old PSX games up.
Link55 said:
Quoted, for your attention.
Also, downloaded the system? The what? You can just download the games and play them as is. Or, in the case of PSOne Classics, save them on your PSP memory card and play them on the go, too.
EDIT- Also, try this handy dandy list [http://en.wikipedia.org/wiki/List_of_PlayStation_3_backward_compatible_PlayStation_2_and_PlayStation_games] to see which games you can play on which model.