PHOENIX DOWN!
I like to mix it up, and use everyone, except Aerith and Cait Sith. I don't like Aerith or Cait Sith in battle because they're best as mages which aren't that useful in FFVII and have suckish HP and vitality. Their limit breaks are really their only good qualities imo, and Cait Sith's are only potentially good. And Aerith's infamous death makes her a bit of a waste of EXP. The only time I really put her in my party if I'm not going for her date and I don't have to put her in is right before Gongaga, just for the plot she gets there.
I have to use Cloud, and even if I didn't I'd always use him as he has the best stats in the game. I think Barret, Tifa, and Cid are the most useful.
Barret has very high strength, and the highest next to Cloud, as well as excellent HP and vitality. He does have low spirit, but since he's a long ranged attacked he's best in the back row where he gets hit less often anyway, so overall he's a great character.
Tifa's got high strength too, and the highest next to Cloud and Barret. However, her defensive stats suck, with both HP and vitality being below average while her spirit is average. She is however, one of the fastest characters in the game. She's a bit of a mix between lightning bruiser and glass cannon. Her limit break's are also cool, and God Hand + Deathblow = win. Not to mention she becomes party leader for part of disc 2, and you wanna make sure she's actually a decent level during that time.
Cid is a very excellent fighter. He's got a good strength stat, only surpassed by Cloud, Barret, and Tifa, and his limit breaks are awesome, and he's got good defensive stats as well. His only real flaw is that he's a tad slow(not in intellect, obviously. In that department he's probably the smartest next to Red). And you'll wanna make sure he's a decent level for when he's made arty leader.
Red is just an average character. Sometimes this can be good, sometimes it can be bad.
Yuffie is below average in most of her stats, but she's the fastest character in the game and has the best luck too. And since she's long range, she's best in the back row, and when you combine the evasion boost from back row along with the evasion boost from luck, and the critical hit boost from long, along with her awesome speed, she's actually a very good character and very ninja-y(will hit the enemy and avoid being hit herself, while moving speedily) in battle. And conformer is awesome.
Vincent's not all that useful either. He's best as a mage, which I've mentioned aren't all that useful. As his weapons are all long range and he is best as a mage, he definitely belongs in the back row, so that's bit of a plus. He's only a good physical fighter if you have the Death penalty, but only if you've killed a lot of enemies with him. If you want good power on your death penalty, you'll practically have to use him non-stop from the time you get him to the time you get the weapon. I generally don't use him too much, but on runs when I do use him I use him a shitload to power up death penalty.
tl;dr Cloud, Barret, Cid, and Tifa and occasionally Yuffie are the way to go.