Final Fantasy VII/Basic combat

General combat strategies
Back attacks: If you're getting attacked at back, there's an easy remedy for that. Hit [PgUp]+[PgDn] (both shoulder buttons on PlayStation, (presumably?) 3 + 9 on PC). Normally this makes your party run away, but with just a brief push, they won't. And after they've tried running away, they always turn to face the enemy. This way, you're never actually back-attacked.

General boss combat strategies
Or, good advice on how not to die in the massive boss battles.

Heal often. How often you need to heal a character depends entirely on how much damage the boss does, but as a general rule, if the character has less than a half of HP left, it's always a good idea to heal up. At first, you probably rely on Potions and Hi-Potions, but in general, using Restore materia is much cheaper and more effective in long run. Once you have the Big Guard enemy skill, White Wind becomes a rather handy enemy skill as well, because Cure2's effectiveness is reduced by Big Guard or MBarrier, while White Wind goes straight through it. Same also goes for Aeris's limit breaks.

Protection and haste. The most useful spells in boss fights are Haste+All, Barrier+All and MBarrier+All. This makes your party faster and gets less damage. But in general, if you have Big Guard enemy skill, things are much easier.

Fury. If the boss is strong against magic and needs good physical beating, using Hypers to get the party to Fury status makes them more likely to get limit breaks.