This is why I skim guides to try to get everything I can out of a game in one go (Hopfully without spoiling myself). I would love to have a website that documents missables in every game without spoilers hahaha.
Usually just typing "X game missables" gives me a link to some gamefaqs post that usually list them. Although of course, I haven't tried it with actual games that much.
Sorry for this wall of text:
As for me, depends on the game. As I can't get bothered with most games that get released today, (Total lack of interest in most cases) I've being replaying games quite a bit since a year or two (thank god there's hundreds of them). Most of the time I just replay it as it is, as the last time I played it was ages
Others I may try to cheat into the game to make it more interesting, like for example, after my first playthrough, each time I want to play Twilight Princess I use a cheat code to multiply the attack power of enemies and hazards, sometimes by two, others by three. It makes for a much more fun ride, and it makes all those special abilties you get even more useful, and it makes far more important to look for extra health or have some type of healing item so those rupies can be used more often. It also makes you want to improve with your bow/wiimote skills, as snipping enemies makes things better when one guy can take three, maybe even four hearts in one attack. Heck, I even use the chain and ball as a defensive tool because of the damage, as it can stop arrows around you while swinging.
Sadly I can't do this with every game. As a silly extra I can change Link's tunic colors as I go through the game for an extra superflous touch.
Other times, I try to play them in a different way. Maybe different characters/builds/classes, for example. For example Super Mario 3D World has 5 different playable characters, and each one is a little different, but with those small things you can play levels a in a different way, so it adds some extra value. And that's just a platformer, other games make each character feel different entirely.
Although lately i've been hacking around games so I can replay them with changes instead of handicapping myself. Mostly RPGs, as they are usually easy and fast to change around to give it a new flavour (Compared to design levels for a platformer or something like that). Not something that anyone would do, of course, but whatever.
Another example would be to use a different weapon, as sometimes it changes how the game is played, like for example Monster Hunter, where each weapon type makes you act in a different way, at least if you want to use them properly.
I played through Resident Evil 4 with only the mine launcher (once you get it), and my god, it was fun. Having to calculate the distance, which enemy should you aim to, so the mine can impact as many enemies as possible, and things like that. I guess it helped that I played the Wii version and the wiimote makes aiming so much more fun/faster, though.
There's also the (Maybe silly) idea that I usually don't 100% my games on the first playthrough so on the second I try all that I didn't touch before, so I have something to look for again.
Some games I never get tired about, like Mole Mania or Donkey Kong 94 for the Game Boy, or Vandal Hearts for the PS1. There's something in them that gets everything right for me and make me want to replay them again and again.
And then there are some that can take thousands of hours out of you. I'm looking at you, Nipon Ichi games
. I would get mad at you, but when you let me make weird things like rabbits being magic overlords wielding old guys as weapons, I just can't.