Absolutely! I know a man who went from a community college to Davidson (top 10 liberal arts in the nation) and a lady who went from community college to Duke (top ten national university).
You will need to make yourself a distinguished candidate, though, as a lot of other transfer applicants...
It depends on your state, but you almost always are required to take a fairly challenging written test (very rarely a road-skills test for permit). Usually you are required to be accompanied by a parent or legal guardian who must sign some forms for you. You often must bring a birth...