What is the best way to plan a 20 day vacation to Europe?

As pointed out by the first answer, the budget is important. I will describe based on a medium budget. These are trips and not a complete itinerary. For complete itinerary you can hire me as your travel planner :) Hostels are cheaper than hotels. Hostels will have private rooms too (in addition to dorms). In Paris, there are budget hotels by Accor (http://www.accorhotels.com/hotel-directory/gb/europe/france/paris/paris-hotel.htm). Traveling by train is cheapest and picturesque. But for long distance you might check cheap flight options, remember that cheaper flights have terrible luggage allowance (or luggage is extra). 20 days is less for so many countries. I will suggest you to stick to France and Italy since your must-to-visit places are in these countries. Add Switzerland and main cities like Brussels and Amsterdam if you have to. Fly in to Paris and out from Rome. Though your flight costs can be more. If you are confident of left-hand drive, you can hire a car in Paris and go around to visit the countryside. This should take 6 days. From Paris got to Brussels by train. From there to Amsterdam. You can spend 2 days each in these cities. 10 days are spent. Then fly to Geneva. Book a Swiss pass and go around Switzerland for 4 days. Traveling by train in Switzerland is highly recommended. Reach Zurich. Mount Titlis and Mount Jungfrau are must, though Mt Jungfrau might be closed. Weather during Sept/Oct is very unpredictable. 15 days are spent. Take a train from Zurich to Milan. From Milan to Pisa and back to Milan. Take a train from Milan to Venice. 17 days are spent. Three days in Venice is must. 20 days are spent. So you are out of days. In that case you can cancel Brussels and Amsterdam, and go directly to Geneva instead. Saves you 4 days. Venice to Rome by train. 4 days should be enough to go around Pompeii and Capri. But for the islands, again the weather might be a problem. Do not go if the forecast is bad, you will not enjoy the islands if it is cold and rainy and more chances are you getting stuck on the islands. Fly back from Rome. You are welcome.

20 days is very less to actually explore all the countries that you have mentioned. Limit yourself to 3-4 countries with at least 5 days in each, if exploring is what you are looking for. Otherwise I would like to recommend you visit the east European countries like Czech Republic, Slovakia, Hungary, Poland, Croatia, Bosnia and Herzegovina. I recently read an article about these countries in outlook traveller and found it very interesting. Apart from being something different from the usual places, they are also more pocket friendly as compared to their western neighbours.