Big Little World and Afrika as well as Home (Home comes free with the PS3.) These are for families.
If you are a shooter person, try Call of Duty 3 and 4. I heard Bully: Scholarship Edition is also good, as is Grand Theft Auto IV for older people.
Playstation store is similar to Wii Shop on Wii, where you can buy Exclusive Games with real money and demos of games you are going to get. You can also do a thing similar to Virtual Console, where it allows you to stream PS1 games to your PSP. The store is for PS3 and PSP games only (not PS2 or PS1.)