We've been on several European cruises and returned from a Scandinavia cruise recently. I highly recommend Princess and suggest that you stay away from Norwegian (it can sometimes be cheaper, but it's a ship that needs refurbishing and there are many other problems). Celebrity and Royal Caribbean also have ships in that area, and they are both good cruise lines. We've been on Royal Caribbean in Europe but not Celebrity in Europe. But I don't think you would be disappointed in Celebrity, and they had great discounts in Europe in 2003. We've also been on Costa and really liked it.
I haven't met a European port that I didn't like, so it's hard for me to even name favorites, but please consider that Istanbul and Malta area also fabulous. Stay in Venice as long as you possibly can. (We've traveled extensively, and Venice is my favorite city anywhere). St. Tropez, France was a great port for us. If you get to the Italian Riviera, Portofino is one of the most charming places that I have ever visited. It's small and is called the "Pearl of the Mediterranean." Other favorites includes Monte Carlo/Nice. Rome is also great, but I would not recommend it for a port stop (pre- or post-cruise would be better, and check out the Hotel Britannia if you do). For the Greek islands, we liked Mykynos (sp??) the best. Athens is great to see the historical sites but is way too big and crowded to get an idyllic Greek island experience. Barcelona is great.
Many ships are sailing the Scandinavia/Russia (Sweden, Denmark, Finland, etc.) with really good prices, and the advantages to sailing to that area are that you can easily see the ports on foot on your own, which greatly increases the quality of your port experience since you're not herded around from place to place wasting time while the bus is being loaded everywhere on the shore excursions, and English is widely spoken. But I guess I would give a recommendation that, depending on your heritage and interests, the Mediterranean would be ideal for a first-time cruise to Europe. With the artchitecture, history, and scenery, you just can't go wrong. If you are of English or Irish heritage, we also really enjoyed our British Isles cruise, and if you go there, you must see the Ring of Kerry. I believe we visited that area from the port of Cork, Ireland (we had 3 ports in Ireland). Ireland is so beautiful, and the people are so friendly and charming. That cruise also went to the Normandy area in France (my husband did a beaches of Normandy excursion while I went with some friends on a day-long trip with a local taxi driver to the villages of Deauville, Treauville, and Honfleur, villages that inspired Monet's paintings) because we had a post-cruise stay in Paris.
If I can help you with more info after you select an itinerary, please feel free to write back. It's so wonderful to visit Europe and very convenient to visit by cruise ship, and I'm glad you'll have the opportunity to go. I hope you have a great time.