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Aluminum polish is an effective way to remove oxidation from aluminum wheels, which aren't well protected from this kind of damage. Use a clean rag to rub aluminum polish onto the wheel, working in the direction of the grain. Work in small sections so that the polish doesn't dry before you work it in. Avoid rubbing in circles as it will lead to a streaky finish. An example aluminum polish is Wolfgang Metallwerk Fine Aluminum Polish, but any polish designed for bare aluminum should do the trick. You can find these kinds of polishes online, at a superstore, or at a mechanics shop. Remember, you want to rub polish over small sections at a time to avoid the polish drying before you rub it in. Move deliberately around the wheel, section by section, and keep track of what you have already polished so you don't end up with an uneven finish. Rub along the grain of the wheel as you go. Polish the lug nuts once you have finished the surface of the wheel. Place a soft cloth around the toothbrush so the bristles don't mar the finish. Use the toothbrush to reach into the spoke areas and rub on polish where you could not reach on your initial sweep. This will also be useful for polishing around the lug nuts. Rub with the grain to dry off the wheel and remove the polish, working around the wheel deliberately like before. Always use new towels at each step in the process so you're not rubbing polish or dirt back in when wiping the wheels clean.
Rub aluminum polish onto a small section of the wheel with a rag. Work your way around the wheel with the aluminum polish section by section. Use a cloth-wrapped toothbrush to polish hard-to-reach spots. Wipe the wheel clean with a soft clean towel.