Personally, I strongly recommend the following:
GRANDIA
My all-time favorite game, hands down, and somewhat of an underrated gem. It has the general atmosphere of a bright and colorful cartoon, while still retaining that epic feel you expect from an RPG. Not to mention, protagonist Justin is cheerful and optimistic! (Yeah, I really hate the moody, emo ball-o-wangst type protagonist you get in so many games these days. I got fed up with those after an hour of Final Fantasy VIII!) Also, Grandia's battle system is widely regarded as a VERY welcome break from the static, turn-based system used in Final Fantasy games of the time.
FINAL FANTASY IX
The Final Fantasy series started out as medieval-style fantasy, gradually evolving into science-fiction as each game would feature a slightly more futuristic world than the last. FF IX was made as a one-off throwback to the series' roots, providing a refreshing return to the green forests and ancient castles of the first few games - only with VASTLY improved graphics! (Yeah, you've probably played this one already, but no list of my personal favorites is complete without it!)
LEGEND OF DRAGOON
Three and a half years in the making, this game provides a fast-paced, intricate story and an eventful journey through an ENORMOUS and richly varied world. (I knew this game was epic as soon as I opened the box. Not only did it come on four discs, like Final Fantasy games of the day - it also had a two-volume manual!) LoD has been accused of borrowing quite heavily from other games, and while this is undoubtedly true, it must be said that it does an excellent job of bringing all the elements together and creating a genuinely unique gameplay experience!
MEGA MAN LEGENDS 1 & 2
OK, so it looks like we won't be seeing the conclusive part of the trilogy any time soon (F**k you too, Capcom), but if you're in the mood for a bit of light-hearted adventuring, you can't go wrong with the first two games. Special mention goes to the recurring villains: The Bonne siblings, a trio of ruthless (but comical) air pirates who have never been bested in battle - until Mega Man shows up... If you ever laughed at Team Rocket, you'll love these guys! They even got their own spin-off game, The Misadventures of Tron Bonne!