Prague and Salzburg are the outliers on the agenda. If you are going to Vienna then Budapest makes the most sense. For the train times add 1.5 hours for getting to and from and a little time to get lost and worry you are standing at the wrong track. For planes, maybe 3.5 hours for to and from the airport and early check in times. For a rental car, maybe 2 hours for pickup and drop off, etc…. then getting back to where you want to be in the town. The shuttles are pretty much door to door.
Prague to Cesky Krumlov 3 hours by shuttle
Cesky Krumlov to Salzburg 4 hours by shuttle
Cesky Krumlov to Vienna 3 hours by car/shuttle, 4 hours by train
Prague to Vienna 4 hours by train
Salzburg to Vienna 2.5 to 3 hours car or train
Salzburg to Budapest 5.5 hours by train
Vienna to Budapest 2.5 hours by train
Prague to Budapest 1.5 hours by plane (Czech Air $125), total trip 5 hours; 7.5 hours by train
The classic Route, almost doable in 10 days:
Depart the US
US to London to Prague (EasyJet, British A. and Smart Wings all have non-stop from Gatwick and or Heathrow for under $200, some under $100. Departure times vary widely by the day of the week and which airport. Making a connection might be interesting)
Prague
Prague
Prague to Vienna (5 hours)
Vienna
Vienna
Vienna to Budapest (2.5 hours)
Budapest
Budapest
Budapest to London to home. (British Air, Norwegian, EasyJet have non-stop service to Gatwick and or Heathrow for under $200. There are morning flights and it’s conceivable you could go all the way home without staying over in London)