Since no one else has said it yet, welcome to flyertalk, UPOD. You may get more specific answers in the Europe forum (under "Travel & Dining: Destination >> The World"). And you'll also get more helpful advice the more refined your plan becomes and the narrower you can make the subject of your inquiries.

Personally, in a 3-week first time to Europe trip, I'd do Rome, Venice, the Alps, and Paris. But that's just my preference.

In understand you want to see as much as possible in the time you have, but you could always wait until you're in your 60s and take that 3-week all-expenses-paid package tour of LondonParisRomeFlorenceVenicePragueAmsterdamVienna StockholmMunichMadridBerlinMilanCopenhagenBarcelon aLucerneEdinburghSalzburgGenevaTheHagueBrusselsBud apestdidItouchallthebases.

In reality, the fact that you're making the trip now, while you're young, and the experience you'll have on the trip you take, not to mention the tricks you'll learn if you stick around here on flyertalk, will insure that you'll be back again and again, and ultimately have time to see all the places you want to see but don't have time for on your first trip.

Bon voyage!