You can play PS1 games on PS3 (either by buying the game digitally or if you have the disc you can insert that).
Now on older models of PS3 you could play PS2 games on them (Model CECHE was the last one that offered PS2 backwards compatibility) so anything after 2007 has no backwards compatibility with PS2 games (if you put the disc in the drive).
They are now offering PS2 games ported to the PS3 and sold in HD collections (Ratchet and Clank, God of War, Sly Cooper, Devil May Cry, Prince of Persia, Metal Gear Solid, etc...) which are good if you missed the series on the first go around.
As much as I looooved some of the games from the PS2, the hardware is such not what I've been babied with in the last few years, hah. Plus, as others have mentioned, I can play my PS1 and PS2 games on my PS3.