I drew this in Illustrator:



Now I'm trying to color it, but I have problems. When I change the fill color of some area, it acts weirdly, because some lines aren't full shapes, but just other shapes connected with the pen tool. (I hope you understand what I mean). So when I change the fill color of something, I get something like this:



Can I somehow join the shapes properly, or is there some kind of tool that colors it like the paint bucket in mspaint would do?

Sort of hard to tell w/o the actual file, so that is not a fully closed path and just the line right? If you're trying to fill it with a color you should complete the path all the around or else will end up w/ the results you are showing. So you would go all the way around like this, if this makes sense. Also you can keep that line and create another layer to make the fill shape below. And to join shapes you can use the pathfinder options depending how you want to join them there are a few options to play with. BTW nice illustration.