It depends, really. Hard mode these days tends to mean either "lol let's give the enemies more health" or "lol let's make enemies do more damage" (or sometimes both) which isn't so much challenging as it is just more time consuming. If I'm going to play on hard, I usually will with my first playthrough, because I haven't learned the game's patterns/mechanics yet and it's actually challenging, rather than just the same game I played before on a lower difficulty but more time consuming. I think the only game that has deployed standard "buff enemy" tactics on higher difficulties that I've found genuinely hard in recent years is Resonance of Fate, and that's mostly because on the standard difficulty that game was brutal. There was also the Forgotten One in Castlevania: Lords of Shadow's second DLC, but we don't talk about that.
But I don't mind starting on higher difficulties, as sometimes games can be dreadfully boring on easier difficulties. Rhythm games in particular really suffer for the lack of notes they have on easy mode, because they're not even remotely in time with the beat, which ironically makes them harder for me as I want to press buttons at different times which make more sense than what the game wants! It also makes a story mode playthrough on a lower difficulty more enjoyable, because it's ridiculously easy by comparison and you can just blast through the whole thing.
That said, I strongly dislike being forced to play on hard mode, because there are always one or two certain moments that will REALLY screw you over, no matter how good you are. Bioware, I'm looking at you. Mass Effect 3 on Insanity and Dragon Age Inquisition on Nightmare are two of the least fun experiences I have ever had and I hate you for it. Timesink games with a lot of story content should NOT make you play on hard, period. If I want to enjoy the story I'm not going to play on hard, and being forced to for a trophy is a serious pain in the backside. I've had some horrific stage experiences in Warriors game as well which have really put me off them at times. Difficulty-related trophies need to die. Like I frequently do when forced to play hard mode for a trophy.
Usually I'll go easy or normal, but sometimes I'll go hard if I'm bored. If trophies are involved, I'll go hard and moan a lot about it if it's actually hard because, of course, that's grossly unfair, haha.