Verified answer

Rome, Italy (also visit the surrounding countryside - it's beautiful)

Southern Germany - the Bavarian Region (it's pretty snowy right now, but it has some really cool stuff)

Paris, France (Eiffel Tower, the Louve Museum)

Amsterdam, Netherlands (the canals & buildings are really interesting - plus they've got some great food!)

Brugge, Belgium (cool old buildings, friendly people, nice atmosphere)

I'm sorry, but you are not the first to ask this question. Do us all a favour and look at the resolved questions first. Better yet, buy a good guidebook and see what this book says. Let's Go, Rough Guide, Lonely Planet are good, check them out. Alternatively head over to wikitravel and wikipedia and start reading there.

http://wikitravel.org/en/Europe

http://en.wikipedia.org/wiki/Schengen_Agreement

http://en.wikipedia.org/wiki/Europe

Personally I like Quedlinburg, Berlin, Salzburg, Prague and Florence.