Ive done this numerous times in numerous ways.
Cleaning them is just a bunch of wet sanding. Keeping them clean can be a few things.
The grit level you start with depends on how bad the lights are. Honestly, being able to get a good result without starting lower than 1000 grit is pretty rare.
Depending on how bad the lights are, you need to start with 400 or 600 grit. It feels like you're ruining them while you do it, but a lot of them need it. There's been several times where i started with 1k and worked my way up only to have to start all over again with 400 or 600 because i didnt like how it turned out.
But basically you are just going 400/600 >1000 > 1500 >2000 > polishing compound > sealer.
Automotive clear coat is probably best. I havent done it. I buy one of the kits (it comes in a foil-looking bag for ~$20) that comes with some pouches of really badass sealer in it. I've found that if you are sparing you can stretch one bag over ~3 sets of lights. Failing either of those, Waxing regularly is the next best thing.
But, I honestly only buy that kit for the sealer. I havent seen ANY kit that comes with the necessary amount/variety of abrasives to do a thorough job on anything more than the most mildly discolored lights. On the plus side, Walmart sells everything you need and almost everyone has one of those nearby, whereas some of the better kits can be tricky to get a hold of locally.