Should I clean the grease off the gears? I usually keep my chain well lubricated. However, my gears are usually greasy. Should I be cleaning the gears or are they supposed to be greasy?

I clean my gearing and whole drive train every few months. I ride about 250km a week so it's not a time frame requirement but a use thing.

I would suggest that every few thousand km or each year you should take apart your whole drive train and get it entirely clean. Replace your chain but keep the old one just in case the new one skips. I replace my chain with each cleaning so that the gearing doesn't wear to fit the chain and thus cause skipping. Well worth the price of a few extra chains vs the cost of a replacement cassette.

My goal for each cleaning is that all parts should be back to their original colour. The older they are the less you can acutally do that as they scratch up but I'm sure you get the idea.