cleaning
There's a couple different ways you can do this, I believe the " black shinny " drums is a wrapped finish.
Id just take a cotton cloth, get it a little damp and whipe it down. Then dry it, should be as good as new. BUT. If you really wanna clean it up, pull off the rims, the heads, then take off the lugs and really give it a good one, just dont get water on the inside of the shell obviously, and make sure as to not use to much water on the outside, get some mystery oil and soak the lugs and such in it over night then whipe the excess off and assemble the drums again, should be lubed and ready to jam. Dont use any bullshit like furniture wax or pledge, or dish washing soap, it's horrid on a drum finish.
If your drums are laquer and the black shinny finish u describe is " piano black" there is a mild laquer cleaner will do.
I can show u a link if u need it. Its like 3 bucks for a bottle.
(ps chrome polish rules on rims and lugs)