This might be a bit regional. I wouldn't say "clear up the garbage' but I know people who do say that.

You would probably say "clean up the garbage". You can also clean up a room, or a house, or I guess even a lake if there was trash in it. When you say "clear up", this means either getting rid of clutter, or making something easier to understand. I could say "clear up the purpose of this presentation", meaning make the purpose of the presentation easier to understand.