I've tried using keyframes, but they do not seem to work.
They do. People use them all the time and I'm 100% positive they are working in your case as well. You're probably just missing something very basic. Anyway, without seeing a screenshot of your timeline/ graph editor and comp probably nobody will be able to help you, so create some and attach them using the camera icon in teh post editor.
Everytime I move a few frames forward and rotate the car, a key frame is not added. When I play back the video, the car stays in the same position where I left off. Could you also explain why the car does not have a solid look to it? It looks a bit ghostly.
Where would I find the graph editor? I'm really new to After Effects.
Well, not meaning to be rude, but have you considered simply reading the help? Seriously, start here. Your screenshot doesn't show any rotation property, so you're not keyframing it. It seems you are just moving a mask around and for that to have any effect, naturally you will have to animate the mask path property.