Unless I missed it (and I haven't skimmed the whole thread), you really haven't told us what you're interested in and that's important to make recommendations.  Are you interested in ancient history?  Medieval history?  Art?  Culture?  Cities? Food?   I think Italy probably has the best overall combination of all of those.  For a first-time trip, I'd also recommend Paris and it is very easy to get around in via cabs, subway, buses, etc.  Ireland is another gem that I can't recommend enough.  Here are a few of the cities I've been to in Europe and some recommendations, in no particular order:
1. Paris - Fabulous city.  If you plan to stay an entire trip there, I'd recommend looking at the Paris Pass to save money on museums and attractions.  I could probably spend a lifetime in the Lourve. Also, for sidetrips, check our Versailles (IIRC, entry is covered by the Paris Pass EXCEPT the admission to the gardens) and maybe do 2-3 nights in Normandy, with tours of the D-Day landing beaches.  Bayeux is an excellent home base in this area and is also home of the medieval Bayeux tapestry, which I found really cool.  Bayeux also has a really awesome cathedral built by contemporaries of William the Conqueror.
2. Barcelona - Fabulous architecture, food, and a beautiful city.  Watch out for pickpockets though, as Barcelona is notorious for it.
3. Rome -  Probably my favorite city of them all, Rome has an incredible mixture of ancient and medieval artifacts and sites, along with great food, friendly people, and a compact center which is easy to get around in.  I'd recommend the Roman Forum, the Vatican museum, and ample time to walk around at night as Rome is really a cool city at night.  Rome is also notorious for pickpockets, so be careful there too.
4. Florence - Florence is a couple of hours from Rome by train and is a fabulous city which just feels medieval.  Check out the Uffizi, the Accademia Gallery, and make sure you have time to walk around and enjoy the city.
5. Munich -  We went to Germany a couple of years ago in the fall and happened upon Munich during Oktoberfest.  We went to Oktoberfest on the closing weekend and it was absolutely crazy, but at least we can say we've gone once.  

  I personally wasn't a fan of Munich the city - it might have been due to the crowds and I'll give it another chance later.  Not far is the town of Fussen, which is a nice little town which is a great home base to see Neushwanstein, Hohenschwangau, and Linderhof castles/palaces.  Be advised that GPS will likely route you through Austria to get to Linderhof and it is an incredible drive through the Alps - be forewarned, though, that you need to stop at a gas station in Germany (PRIOR to crossing the border) and purchase the permit for driving in Austria and stick the sticker on your window.  The Austrian cops sit at the border and pull people over who don't have this and it is a pretty stiff fine.  I'd definitely recommend getting a car though - a pro tip to save money is that you don't want to pick up a rental car at an airport or train station, as IIRC, there is an insane surcharge (30% or something like that) associated with it.  When we went, we flew into Frankfurt, took a train to Cologne to visit friends, and then picked up a car at a rental place in downtown Cologne.  We then returned it in downtown Munich and took a train to Salzburg and continued our adventures.  

6. Cologne -  Cologne is a nice town with a famous cathedral, a unique beer (Kolsch), and a scenic walk along the Rhine.  We only stayed here one or two nights and then grabbed a car and started our trip through Germany.  I highly recommend going to a town like Baccarach and taking a short cruise up to St. Goar and back, as you'll see a ton of cool castles along the way.  We then went to Rothenburg, which is a medieval walled city and one of my favorites.
7. Vienna - I'd say Vienna reminded me a lot of a smaller version of Paris.  Great food, interesting sites, and lots of free musical performances to listen to.
8. Salzburg - Gorgeous city and one of my favorites.  There is a huge fortress overlooking the city (Hohensalzburg Castle) which is cool to visit and lots of interesting sites.  The Sound of Music tours originate here and take you through some incredible mountain scenery, which is worth it alone IMO even if you don't care about the movie.
9. Dublin -  I'll be honest - I didn't care for Dublin (or Belfast).  The key to Ireland is the small towns and rural areas.  I normally don't recommend spending entire vacations on a tour (day tours are fine though), but we used Vagabond tours (small, 12-15 people tours) and saw the entire island.  It was worth every single penny IMO
You also just missed a ton of great deals on airfare (some might be going on still for all I know).  For example, we're going to London in March ($550 round trip from Chicago and that, unfortunately was booked BEFORE the great deals came), Barcelona in May ($400 round trip from Indy), and Rome in October ($420 round trip from Indy).  I told a buddy about them and he got round trip airfare from Denver to Madrid for around $400 and it went up to $932 literally hours later.