A long adventure full of wit, action, and classic characters, this is an 'old skool' RPG that needs a little more zing.
BoF III is a very long and satisfying RPG in the traditional JRPG style -- turn based combat, a set of silly but fun and interesting characters, anime style artwork that reminds you of a GBA game rather than anything more modern in style, minimal character customization, a long and very typical storyline. If you long for the mid-90's style JRPGs like the early Final Fantasy games, somewhat tactical turn based combat, cute and funny characters, and a lightweight story, then this is a great choice for you. The main game is quite long (40-60 hours, depending on how quick you take it).
That said, if you've played and enjoyed newer RPGs, this game will be a lot of fun for about 10-20 hours for you, then start to wear. Random encounters can be skipped, but are largely necessary to help you "level grind" and be ready for boss fights in the main story. Quests are pretty basic "go kill this stuff" in nature, though they do mix it up a little with some semi-stealth sequences, and a few dialog-tree related puzzles. Minigames are pretty limited as well, with the most sophisticated being the fishing, which is almost a separate game in itself. But I found about 15 hours in that I WANTED to finish the game, but was tiring of the "sameness" of everything. There's not a lot of variety in visuals, gameplay, questing, or combat. So you'll either get addicted and play through it, or you'll burn out and move on to something else.
Visuals and Sound:
Breath of Fire III is considered a classic RPG on PSOne, and was released in Europe and Japan for PSP. The PSP version is a slightly polished version of the PSOne release, but retains basically the same visuals and music. These would be "typical" visuals on a DS title, but for a PSP game they're pretty dated -- they're similar to the Final Fantasy I and II releases on PSP, rather than some of the more stunning recent titles you've seen like Final Fantasy VII or Disgaea. There's no spoken dialog, and the music is well done but definitely has that midi sound of many PSOne titles. The characters and environments all use very tiny artwork, which detracts from the immersion factor a lot.
Overall, a recommend if you long for a great old-skool RPG, or a skip if you've been spoiled by the newer and more nicely-produced RPGs.