Where are some of the best places to go on a European vacation?

I have never been out of the country I am interested in possibly planning a trip to Europe. I'd be traveling with one other person. I'd like to pick an area where we can take a bus or train to get to surrounding areas. Accommodations do not need to be fancy-- just a young couple exploring new places. Have any of you traveled around Europe? What are the countries I need to go to and the places that I cannot miss???