All musical instruments have their own cleaning quirks. Some are easy to store away so they can be kept free from dust. Some need polishing. Some just need dusting. The wind instruments need to get all the gunk out of their insides with something like a large pipe cleaner (I don't know if I want to know how that got in there and what it is. Somebody puts something full of that muck in their mouth?). Drum kits, of course have their own quirks.
The quick and easy solution to keep your drum kit nice and shiny in concert-worthy dust-free perfection is to keep dust covers over it at all times. If your drum kit is kept at home for practice rather than in a studio, you won't need to take these covers off, even for practice. Cloth covers muffle the volume of a drum kit, meaning you won't deafen your family, wake the neighbours or generally make a nuisance of yourself. However, you can't do this for the cymbals (crash or hi-hat) very easily. And you may not want to keep the covers on!
The best means of keeping your drum kit clean and shiny is through regular dusting. This does pretty well for general cleaning and provided that you don't try drinking and drumming at the same time or spill things on it, this is all that's needed. You can even ask your regular domestic cleaner to do that for you. The drum kit in my household is about five years old and it is still as shiny as the day we bought it.
If your drum gets a little bit grubby, the most likely place to find this is on the skins of the drums, as this is in contact with sticks (and brushes) that aren't always super-clean. What you do not want to do is to spray a very harsh chemical onto them. While the plastic of the skins is tough for obvious reasons, some chemicals can react with them and ruin the skins - and they're not cheap to replace. Instead, wipe them down with a soft cloth dampened with dilute vinegar or even dilute vodka. This will remove most of the grime. Just water may be enough. Never polish the skins, and be careful if you are shining up the rest of the drum not to get any polish onto the skins.
Never get a leather or hide drum wet. Leather stretches when wet and you will lose the tone of your drum.
Cymbals can be polished with regular metal polish, or else with a paste made of salt and vinegar (no, not your leftover potato crisps!). Do not work in a circular motion, but rub as if you were moving along a spoke from the centre of the cymbal to the outside. Finish by buffing with a soft dry cloth. Gongs can be cleaned in the same way. This is suitable for brass or for copper.
Do not use water on any wooden percussion instruments (e.g. wood blocks), as this will soften the wood and make it more vulnerable to dents and damage, besides altering the note. Use furniture polish, either the spray-on or the pour-on kind, but don't overdo it. You can also use linseed oil as a polish, but once again, avoid being heavy-handed.