Removing oxidation from wheels

It's about time I tackle this project. Curious to know if anybody here has done this and what did you use and what process did you use?

My wheels are not chrome. They are aluminum alloy I suppose, with a portion of them that have that turned machined look and they have some black painted areas on them, could be powder coat I suppose. Regardless, they have some oxidation built up I'd like to remove. I've read some things where people have used Rain-X, toothpaste, metal polish, etc. I don't mind putting in the hard work and effort but I'd like to not go through 3 or 4 different trials to get it.

Also, my headlight lamps are starting to oxidize as well so if any tips for those, that would be great too.

Many thanks.