Check the websites of British, Virgin, Lufthanza, Swiss, Austrian, among others. You may find ads for travel specials. You may also find reasonable priced packages for lodging and ground transportation from those carriers. You will probably want to use the rail system a fair amount, as it is much better and on-time than what most of the US has, so get familiar with Eurail (Britrail, DB, SBB, etc.).

Also, this discussion is probably better suited for the Western Europe travel forum (a sub-head under Travel&Dining). If you search/skim that forum, you will find a number of these type discussions and should be able to get a lot of ideas, as well as recommendations as to where to visit for how long.

Welcome.