I've been to Europe more times that I can count and done a lot of travelling there (I used to live in Norway). I think the best places to go depend on what you like. Since its your first time you probably want to take in the sights. London, Paris, Rome, Florence, and Venice are good for this. If you're into the party/nightclub scene good places to go are Barelona, Amsterdamn, Ibiza (an island that is nothing but nightclubs), and some of the Greek Islands like Mykanos. I, personally, like to relax, eat good food, meet cool people, see a couple of sights, that sort of thing. Cities that I had a great time in were Istanbul, Prague, Dublin, Lisbon, Naxos (the Greek Island). The food was amazing all throughout Greece -- probably my favourite place in Europe for food.
With 3 weeks you obviously have to pay some attention to geography and can't wander around to all the different corner of Europe but something like London, Paris, Barcelona, Nice/Monte Carlo, Rome, Venice, Munich, Amsterdam is feasible if you don't mind staying 2-3 days in each city and travelling over night (if you travel overnight pay the extra 20 euros for a sleeper car, trust me). It may feel a bit rushed though, but you'll get to see a lot. For a slower pace, don't be afraid to stop at the small cities and towns along the way. Although, there isn't as much to do, you'll get to see a few things that are "off the beaten path" and get a better idea of how people in the country actually live. I had a great time staying in small towns all over Germany.
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