Europe's Best Three Weeks
This far-reaching itinerary makes most sense by train; a Eurail Global pass for 21 consecutive days is a good fit. Taking two night trains avoids two long travel days. The day trips from Munich to Salzburg and from Florence to Siena keep the hotel situation streamlined and take advantage of excellent public transportation services. With your railpass, the Rhine cruise is covered, and you get a discount on the Romantic Road bus that runs Frankfurt–Rothenburg–Füssen–Munich (though the train is preferable, entirely covered, and faster). While the route can be done by car (with certain adjustments), this plan is heavy on big cities, where cars are worthless and expensive to park.
Day 1: Arrive at Amsterdam's Schiphol airport, and stay in the nearby town of Haarlem (2 nights).
Day 2: Using Haarlem as a home base, day-trip to Amsterdam.
Day 3: Cross the German border to the Rhine River Valley; Explore the quaint riverside town of Bacharach (2 nights).
Day 4: Cruise the best stretch of the Rhine (between Bacharach and St. Goar, then tour St. Goar's Rheinfels Castle. Return to Bacharach by train.
Day 5: Take the train to Rothenburg and sightsee (1 night).
Day 6: After a morning in Rothenburg, take the Romantic Road bus to Munich (1 night).
Day 7: Spend the day in Munich, or consider day trips: a bus tour to "Mad" King Ludwig's fairy-tale castles, or a train from Munich to Salzburg (1.5 hours each way). Take a night train to Venice.
Day 8: All day to enjoy Venice (1 night).
Day 9: Head from Venice to Florence (3 nights).
Day 10: Spend the day in Florence's museums.
Day 11: Using Florence as a home base, side-trip to Siena.
Day 12: Head for Rome (3 nights).
Day 13: Sightsee Rome.
Day 14: More time in Rome.
Day 15: Take the train to the Cinque Terre and set up in the village of Vernazza (2 nights).
Day 16: Today's a "vacation from your vacation" in the Cinque Terre. Hit the beach or hike the Riviera trails.
Day 17: Train into the Swiss Alps, and sleep in the mountain town of Gimmelwald (3 nights).
Day 18: Alps Appreciation Day: Spend the day enjoying the hiking and high-mountain scenery.
Day 19: More time in the Alps. If the weather's bad, side-trip to Bern or Luzern.
Day 20: Take a morning train from Interlaken to Paris (3 nights), then spend the afternoon sightseeing.
Day 21: All day for Paris.
Day 22: More time in Paris, maybe with a day trip to Versailles.
Whether you have a week — or longer — here are my recommended priorities. These itineraries are fast-paced but do-able by car or train and each allows about two nights in each spot (I've taken geographical proximity into account). Most work best if you fly "open-jaw" (into one city and out of another).
5 days: Paris, Swiss Alps
7 days, add: London
10 days, add: Rome
14 days, add: Rhine, Amsterdam, Haarlem
18 days, add: Venice, Florence
24 days, add: Cinque Terre, Rothenburg, Bavaria
31 days, add: Bath, Salzburg, Hallstatt, Nice
37 days, add: Provence, Barcelona, Madrid
43 days, add: Vienna, Prague, Berlin
45 days, add: Bruges