Well Brian, it depends where and how grotty the part is.
For instance, the levers were like top one in the picture originally.
First of all I spread Autosol thinly over it.
Then I use a dremel-type tool with a rubber circular attachment.
You only get one or two in the case with the rotary tool.
So I bought a pack of 10 of them.
This gets rid of the oxidation, but leaves streaky marks
More Autosol and green scotch-brite to smooth it.
There`s 3 grades of this finishing stuff.
Red is coarse, green is medium and grey is fine.
If the green leaves scratches, use the fine grey stuff. (with Autosol.)
When its suitable for polishing, I generally use a cloth first with a good covering of Autosol, giving it a good hard rub
Then the rotary tool with a small buffer tool and autosol.
To finish, a soft polishing cloth (duster).
If a part is very scratched I`d start with about 800 grit wet & dry.
Then 1500 grit and continue as above.
With covers I buff with a drill fitted with 3 inch buffer tool.
This is a dirty job, ally is black.