I actually won both battles on my first go. I think that's my problem with the system, on normal at least. Feels like you can be shit but as long as you have healing spells and potions you'll still be fine. I assume that's not the case on hard!
You're right that in most boss fights you're 'supposed' to take damage and work in effective heals from Aerith/Barrett during fights.
It does take a while to get used to the intricacies of a surprisingly deep combat system, particularly the best ways to minimise damage.
A lot of it revolves around knowing what enemies are weak to and learning the correct reaction to their moves so you can choose when to counter, dodge, interrupt or block. Hard mode especially.
For Cloud, the best way to avoid damage is to parry. Blocking a physical attack whilst in punisher mode will take reduced damage and launch a counter hit. A nice tip with this- switching in to punisher mode also gives you a counter stance, so if you see a physical attack coming you can tap triangle to switch to punisher and counter it without having to wait until you can switch and block. Towards the end of the game you get a sword with a really powerful counter move that even counters magic and projectiles which takes his counter game to another level. Before then you'll have to block vs some ranged and magic spells and eat some damage.
For Tifa, her speed can get her away from most damage, especially when you give her that pink materia that turns her dodge into a stupidly quick roll (I think it's deadly dodge?) When you have that equipped you can be sprinting and sliding around the battlefield like a demented longjumper and can basically hold down the block button whilst inputting her combos when up close and personal as an insurance policy in case they power through your moves.
Barrett and Aerith are too slow and don't have reliable counters or dodges, so blocking and keeping them healed up is the name of the game. At least Barrett has thick skin and lifesaver so can be a useful tank. Aerith has even less options and feels less fun to play in my opinion as a result.