I've just found out there's also a Sega Saturn version
There are actually two Saturn versions, I believe. One for the regular Saturn, one for if you have that MPEG thingy that goes into the slot behind that door where the save battery is located. But it's all moot unless you know Japanese, as that one stayed in Japan. I think there's also a PC version in Japan, but not sure about that.
As for the rest, SegaCD or PSX are the best bet. There are some changes in the story and some game mechanics between the two, and some people greatly prefer the SCD original, while others appreciate the better sound and visuals of the PSX. Only the SCD has random battles, so if you hate those go with PSX, but other than the world map (where there are no battles at all except on the SCD) it's not exactly easy to avoid the enemies so to me it doesn't really matter about the random battles or not. Either way you'll be fighting a lot in the dungeons.
GBA is very cut down, don't even bother with it except as a novelty.
PSP is a worse translation (they couldn't use the older translations due to rights issues) with worse voice acting for the most part. The new prologue scene in particular has some really horrendous voice acting. It's not as cut down as the GBA version, but still things aren't quite how they were before, dungeons seem way shorter/easier. Going back to the new prologue, it's really odd and doesn't really fit the tone of the game anyway, so I say skip the PSP version unless it's all you can find for what you're willing to pay. You can always convert the PSX version to play on the PSP if you really want to play on the go.
That's just the first Lunar. The second one, Eternal Blue, hasn't been redone since the PSX. Again, PSX or SegaCD, some like one over the other but they're both good. There's some problems getting started in the SCD version, since you have to pay a type of magic experience to save your game, and it's in short supply at first, but after an hour or two it stops to really matter at all. The whole magic experience system is gone in the PSX version, and you can just save anywhere. The changes to EB aren't as drastic as in the first game, but again, some just prefer the original.
One final note, stay far away from Dragon Song, the Lunar game on the DS. IT IS HORRIBLE.