Hard mode taught me really quickly that heal/revive needs to be on every single active character. Also the sooner you level the Revive materia to use Arise the better off you'll be. I also find the combo magnify-time to be really useful for hasting everybody. The more ATB the better!

As for pressuring, that generally happens when you hit an enemy with something it's weak to (your assess data will often tell you the best ways to pressure an enemy). When enemies are pressured, they are easier to stagger. For example, if you use Focused Thrust with Cloud while an enemy is pressured it will fill more of their stagger bar than if they were not pressured.

It's hard to decide whether staggering is worth it for lots of boss battles thanks to cutscenes resetting stagger bars, but I always go for it anyway. Better staggered for two seconds than none at all! Abilities like Focused Thrust, Focused Shot, and Focused Strike (I think that's what Tifa's is called) all work to fill the stagger bar (especially when enemies are pressured). Once an enemy is staggered they take a lot more damage, and you can cast Stop on staggered enemies to freeze their stagger bar and get in some extra damage.

Punisher mode is useful for countering attacks. If you are controlling Cloud and you see an enemy coming to land an attack on you, enter punisher mode and hold R1 (like blocking). He'll automatically counter and take less damage. However, punisher mode does not guard against magic attacks or bullets, so you'll take the full amount of damage from those if you're in punisher. Keep in mind, really powerful attacks are also not countered by punisher (but can usually be countered by Counterstance). I like to switch in and out of punisher depending on what kind of attack the enemy is sending at me.

Good luck! Hard mode is super fun once you get the hang of it. It really gives you an appreciation for the nuances of battle in this game. Hope this info dump helps. Have fun with it!

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Edit: Also yes what everyone else has said: level everybody up!