I think starting equipment, limit breaks, and statistics can help you on that, although their differences in the last are not as diverse.
Tifa and Yuffie are obviously a monk and ninja, respectively. I think of Barret as like the Warrior from FFIII, he has high Vitality (DEF) and Attack, and can used ranged and close-range weapons. Cloud I see as a magic knight/black mage & warrior combo since he has the highest magic other than Aeris, starts with Lightning and Ice, but also has high attack with a sword. I've always though of his spikey blonde hair replacing the pointy yellow hats, and the blue outfit.
Cait Sith represents his own "toysaurus" class, with manipulate, hypno crown, and status materia like transform and seal. Aeris is obviously the White Mage, Cid a Dragoon, and I just make Vincent and Red XIII red mages since they're well suited for it, statistically speaking, and look the part as well (Vincent in outfit, Red in color and in name).
Obviously you can get through FFVII ignoring all this but that's how I play it.