Cloud, Barret and Cid. It's the most powerful team. Trust me. I played the game for years before I came to this conclusion. Here's a brief overview on my character opinions:
Cloud - Insane physical and magical power, with a level 4 limit break that's better than Knights of the Round. The best character in the game, no doubt.
Barret - Insane physical power, and if you know how to use him right and how to power up his Missing Score, he's irreplacable. Most of his limit breaks kick ***, especially his level 4 limit break.
Tifa - Good frontline fighter, but her 'chain' limit breaks don't give her a position in your main party. Just use her when you have to and ditch her otherwise.
Aeris - Has magical power that defeats Cloud's and her defensive limit breaks are great, but she kicks the bucket halfway through the game, sadly, so you can't use her awesome level 4 limit break in the tougher battles later on. A good unit to use throughout the first disk, though.
Red XIII - Has well balanced physical and magical strengths, but a poor level 4 limit break makes him lose points.
Yuffie - Performs slightly better on the physical side and slightly worse on the magical side than Red XIII, and also loses points due to a poor level 4 limit break.
Cait Sith - Average physical and magical powers and his limit breaks suck about 95% of the time (the 5% is when you either get lucky on Dice or get the 'Death to All Enemies' roll on Slots). Reeve could have controlled something better...Cait Sith is the worst unit.
Vincent - Overrated. His limit breaks are great for the majority of the game but their usefulness declines towards the end of the game. Still a good unit, with great magical strength.
Cid - Cid is basically Cloud with stats that are not quite as high, and has some of the best limit breaks in the game, especially his level 4 limit break. One of the best characters overall.
Sig and avatar by Ozzo. Thanks.