Traveling is my passion. I lived in Rome for a while, so pm me if you really need advise to see hidden treasures of Rome (believe me, it is more that the tours and books tell you).
Depends on what you would like to see, touring Europe could be tricky. You have to decide what you want to see/to do there: typical tourist stuff (riding Venezian gondola, top of Eiffel Tower, pictures with Coloseum, Loch Ness), cultural stuff (weiner eating in Munich, sailing in Amalfi Coast/Mediteranean, "coffee shopping" in Red Light Distric of Amsterdam, eating at cafe along the Seine), outdoor recreation (paragliding in Interlaken, hiking in Cinque Terre, ice climbing in Jungfrau), or art seeing (Accademia in Venice, la Giaconda in Louvre, British Museum, Musei Vaticani, Musee d'Orsay). Or maybe the Nazi camp tour (Dachau, Auschwitz), Christian tour (Vaticani, Roman catacombs, Asisi) or architectural tour (Familia Sangrada in Barcelona, Fred Ginger in Prague, Pampidou in Paris).
It seems that you will be in Europe early June, that says a lot. If you are happen to be in Rome for the Pantecost, you HAVE to attend the 11 am mass at the Pantheon. They have annual red rose petal droping from the Occulus (the top opening of the Pantheon). Once in a lifetime opportunity to see this touching moment (I've heard so many people cry because it is so beautiful)
I missed it this last year, since I chosed to attend the Venesian choirs for Pantecost. In every corner of the city, there are groups of people singing, and the peak is at 6 pm at Piazza San Marco, where they gather (more than 300 people) and sing. It ends where several honor Venesian guards march into the piazza to lower down three flags in front of San Marco, putting them in three huge chests. We're talking about maybe 30 by 10.
Good luck planning your trip.