What exactly did the rings "do" to the wearers?
it's not really super clear, but the elven rings give mastery over water, fire and air (gandalf's ring is fire, galadriel's ring is water and elrond's ring is air... though i may have those last two mixed up). they also, from what i understand, everything that you can do, with the rings you can do them better, significantly so. galadriel very often used her ring to spy on sauron mentally and he was unable to even notice she was doing it, though it was a massive strain on her. actually, i could be wrong that sauron didn't know she was doing it, but either way he was unable to stop her from doing it as well as unable to find her at all. so it was SORT of like a palantir in that regard. they also helped the elves maintain their own lands. after the one ring is destroyed, all the rings became inert basically, and rivendell and lothlorien started to fall apart and decay pretty rapidly. the elves weren't meant to stick around middle-earth for so long but a lot of them chose to stick around in order to be a bulwark against sauron. the rings helped those areas (and the elves within those lands) stay long past their expiration date.
the 9 rings for humans gave them significantly extended lifespans (could have been all the rings did that?) as well as basically extremely good luck for lack of a better description. things just worked out for the people wearing them.
the dwarven rings are... it's really hard to tell what exactly they did because there's barely anything written about them other than the rings didn't work right on them. also, initially sauron wanted all the rings to go to elves, but celebrimbor righteously screwed that plan up. so i think it's assumed that they would all have the same effect (besides the 3 elven rings), but either way the 7 dwarven rings absolutely didn't work right, specifically because dwarves fundamentally different than most every other creature.
illuvatar (basically the christian God) made the races of arda (the planet middle earth is on). aule (the smith valar... hmm, okay without getting too nerdy, [too late] illuvatar made 14 valar, who are like gods, small g, and each one of them embodied a specific trait of illuvatar. aule embodied illuvatar's craftsmanship and desire to create things) wanted to create a race too and made dwarves, but he did it without talking to illuvatar about it. so illuvatar confronted him and aule was like, whoops sorry dad, and went to smash them (they weren't "alive" yet) but illuvatar was like, no no it's fine, i'll give them the flame imperishable (basically souls) and adopt them as one of my children, they'll be good kids, but just understand that because you did all this without talking to me first, there will be times when they are at odds with the world around them (specifically elves, but that's a different topic).
SO. it's never actually made clear but i THINK because of that little line of dialogue, when the dwarves get the rings they just sort of... screw off and each clan (7 clans, it's assumed the head of the clan each got a ring, but... well there's conflicting issues with that... but again a different topic) became extreme isolationists. they became super paranoid and just stuck to their own mountains and dug and amassed treasure. the hoard that smaug collected was one or several of the dwarven hoards.
tldr:
outside of the 3 elven rings, the rings gave the wearers long life, good luck and enhanced the wearer's natural abilities... possibly more, but i don't think it's ever said specifically what they do