Each European country is so culturally diverse that you shouldn't base your trip on trying to get to the most countries. People who go on 2 week trips to Europe and visit 7 or 8 countries are wasting their time. Unless you are going for a month or longer, you should try to focus on visiting just two or three countries. The trick is to see as much as you can on your trip without sacrificing the experience that each place has to offer. Luckily with bullet trains and budget airlines, you aren't limited to one region. If this is your first trip to Europe, then I would suggest going to Italy and Germany. You can start in Berlin and ride the train down to Rome, stopping at all of the various cities along the way. If you have enough time you could even include a few days in Prague. (e.g. Berlin -> Dresden -> Prague -> Munich -> Venice -> Florence -> Rome).