I have to mention the original Grandia. Cartoony JRPG, light-hearted but HUGE, and surprisingly immersive. The music is amazing, and the battle system is still hailed as pure genius. This is my personal favourite game of all time.
Also, Legend of Dragoon. Somewhat more serious JRPG, with plot twists out of a soap opera and a rich, diverse cast of playable characters. Some parts of this game will feel more technologically advanced than others, the result of a development process spanning more than three years. It has also recieved some criticism for "borrowing" heavily from similar games. As a whole, however, LoD stands proudly on its own, providing a unique journey through a unique world - and with four discs, you'll get a LOT of game for your money!
Platform-wise, I hereby support everybody who has suggested the Spyro trilogy before me. This is platforming bliss, as good as it gets. DEFINITELY buy the Spyro trilogy.
As for Crash Bandicoot: You really can't go far wrong. Crash and Spyro were pretty much tied for the position of "official PlayStation mascot" for most of the PS1 era, and I don't think there ever was a clear winner. While I personally prefer Spyro, for no reason I can really put my finger on, I have to say that "Crash 3: Warped" is an absolutely fantastic game which anybody with a PS1 and a pulse needs to own.