My original first destination is Venice not Munich (been there, done that). Seems like the cheapest way to get from DC to Venice is to land in Munich and take a budget airline to Venice.
Are the savings enough to compensate for the inconvenience and the extra time? It's always chancy to plan to take a flight on a separate ticket on the same day, especially on a budget airline and especially with the possibility that either airline changes its schedule on you. So it's probably staying at a hotel overnight at the Munich airport or schlepping into the city and back to the airport with all your luggage. Have you considered flying into Milan (sometimes cheaper and there are frequent high-speed trains to Venice)? Or into Rome, then flying from Venice to Budapest?
Anyway, I think you're trying to see too many places in too little time. How many nights were you in Berlin and Prague? Were you there 3 nights each? If so, there's more to do. If you were there longer and you feel like there's more to do, then surely 3 nights in these new cities won't be enough.
From Vienna to Prague the train is okay, but to Berlin, you should fly. From Berlin, the train is okay to Amsterdam, but from Prague, you should fly.
Except for the two moves in Italy, they are going to use up more than half a day each, leaving you little more than 2 full days in each city. And even in Italy, you'll lose nearly 1/2 day moving (packing/unpacking, getting to/from the train station, getting oriented to a new city). Instead of wondering if there's enough to see in 3 days, figure out if the time and cost is worth moving and having to see everything in only 2 days.
