Chrono Cross - despite having not as strong of a cast as Chrono Trigger, this game is absolutely fantastic.
Final Fantasy IX - The best of the PSone Final Fantasies. Note that FF7 is incredibly tough to play today due to its total lack of analog control. You never think about it until you try and play a 3D game with digital control only.
Xenogears - I'm personally not a big fan of Xenogears as a game. There's way too much tedious exposition, but this is one that's looked at as a necessity.
Front Mission 3 - If you like SRPGs, this is probably the best PSone classic available for that itch.
John's assessment here is quite solid. Plus, he said FFIX was the best on PSone, which is the best way to make me agree with anything.
I'd also highly recommend FF Tactics-- although the translation and new content in the PSP version does make that one a little better.
I really enjoyed Xenogears' story and music, but I absolutely hated PLAYING that game. Some contend that the second disc sucks because it's all exposition with little gameplay, but I actually loved it because I got to actually stop PLAYING the game.