Places:
Italy is a must, particularly the smaller towns. I spent three days about 40 miles north of Turin in a town of less than 900 people set in the foot of the Alps. It was the most beautiful and best three days of my summer.
Visit Berlin. Forget Frankfurt. Berlin is a cultural mecca. It has beautiful architecture, a wonderful cafe life, and great night life. If you're in Berlin you can also visit Potsdam, which is the location of the Prussian Imperial palaces. Not nearly as gaudy as Versailles, they are still amazingly beautiful.
Other locations you should strong consider: Warsaw, Cracow, Prague, Normandy (only if you can rent a car).
Aside from that, I would recommend NOT trying to be in too many places. I hate when people say "oh, I've done Madrid" or "I've done Paris". You haven't. If you haven't spent at least a few days in a city, you've only had a taste of what it offers. The longer you spend, the more you discover. Therefore, don't be overambitious in setting an itinerary. When I traveled in Europe after my studying there, I spent a week in Paris and I wished I had more time.
As for travel, buy a rail pass. I would recommend trying to plan out as much as you can in advance. That way you don't need to worry about full trains, which happens more often than you would think.
Buy a backpack. Don't mess around with suitcases. I found my bag for $80 at REI and it saved me a lot of stress and trouble. Also, don't over-pack. That means don't bring a laptop, don't bring ten t-shirts, three pairs of pants, three pairs of shorts, and five pairs of shoes. You'll regret it. You want to bring as little as possible. Along the way I promise you will end up buying a shirt or two.