In my time with the game I finished easy mode first. Hard mode is unlocked after completing the story once on any difficulty. Before doing hard mode though, I'd recommend going to chapter select and choose chapter 16 on easy, to grind materia levelling and character. levels. Equip Cloud with fire and magnify materia and speed run through this section, until you're just about to enter the Shinra building. Equip Tifa an Barrett with 2x HP up and 2xMP and any other materia that needs levelling. Do this for around an hour and you should have near enough everything at max level.
You can then choose to go to chapter 17 and do the battle simulator if you've completed all of Corneo's colosseum (aeirith chapter 9 straight after hell house, Tifa and Barrett chapter 14, Cloud chapter 9 or 14). Doing the last challenge will unlock the ultimate weapon trophy and it will give you the strongest item in the game, which will start you with a full limit bar at the beginning of each battle, and it will gradually refill there after. This will make hard mode considerably easier. If you don't want to do this and just play the game the way it was intended to be played just start at the beginning of chapter 1.
Hard isn't impossible and some fights are very straight forward. Just use elemental weakness's whenever you can and avoid using MP unless you absolutely need to, as MP doesn't replenish when you rest at a bench and all item use is disabled.
Hell House, Eligor and The Arsenal where the hardest fights in the whole game for me. If you're struggling on a particular boss please use a guide or watch a video on YouTube. You may be able to afford to button mash on the lower difficulties, but hard mode is a completely different experience.
Take your time and enjoy this fantastic game!