I'm trying to make my Ginjirou more magically potent, and I know that generally upping both INT and MEN should be just fine, but what exactly does each stat do? Can anyone explain what advantages, say, AGI and LUC have on the game itself?

INT determines how powerful the magic is and the size of the spell while MEN determines how much MP you have and how quickly it's recovered.  So for, say, Randy, you could get away with putting a big chunk of your points into INT, but quite a lot less into MEN, but for nicole you'd need a lot of MEN since she doesn't have Randy's attack capability.  If you don't want to sacrifice too much of Ginjiro's attack power and speed then you'll probably need to choose between having him be able to let off a powerful spell every so often to finish off enemies from a distance (high INT) or having weaker spells which you can fire off rapidly to keep enemies at bay (MEN).
AGI ups your attack and movement speed, but it also decreases the time it takes to cast (and recover from casting) spells, leaving you with fewer moments of vulnerability while casting, so it'll still be important for a magic-using Ginjiro.
LUC is a kind of modifier stat, it increases your chance of landing more powerful hits and your chance of defending against enemy attack.  Nicole also needs high amounts of LUC since it determines which spells she can cast.  You can get by without upping LUC for everyone other than Nicole though.

Hope that helps.