Budapest...then Krackow, Vienna or Prague??

There will be a new direct flight service between Budapest and Kraków (and vice versa) on LOT Airlines beginning in April.
http://www.krakowairport.pl/en/airport,c94/news,c120/lot-chooses-krakow-new-connection-to-budapest-and-more-flights-to-chicago,a2799.html

Most cities in Central/Eastern Europe are well connected by train or inexpensive long-distance bus services (FlixBus, Poski Bus, etc), so you need to look at those schedules.

You say you're spending ten days in central Europe, including one week in Budapest and "the other week" elsewhere. Is it ten days or two weeks?

If it's ten days, and you'll spend seven of them in Budapest, then I'd suggest Vienna for the remaining time because Vienna is closest and easiest to reach from Budapest. Besides the hydrofoil there is fast train service. Vienna has more than enough to see in the time you'll have.

If it's two weeks, then you have a whole second week for any of the three cities, and it comes down to what interests you most. There's enough to see for a week in Prague or Vienna, with maybe a day trip or two. Krakow has an outstanding medieval center, castle, and Jewish quarter, plus Auschwitz and the salt mine nearby, but I'd be hard pressed to spend a whole week there.

Whichever second city you choose, I hope you can fly "open jaw," flying into Budapest and home from your second city. That will save you time and expense backtracking to Budapest.

Vienna is close enough to Budapest. Save Prague and Krakow for another trip.

Clarification - the trip is 10 days, which means 4 days in Budapest and 4 days in either Vienna, Prague or Krackow. Thanks!

It's Kraków (not Krackow), and 4 days would be perfect there because it's just big enough that you can see all of it plus some obvious side trips. Just find a cheap flight from Budapest once they're available (after April 28).

Your city choice should depend mostly on your particular interests and what you want to see most in these cities.

Thanks everyone for your help. We chose 5 days in Budapest and 5 days in Prague. Any suggestions on travelling between Budapest and Prague?

Direct daytime trains take 6.5 hours, about $75 per person in 2nd class if you buy there on short notice. Some advance-discount tickets are also sold on the Hungarian and Czech web sites; see https://www.seat61.com/trains-and-routes/prague-to-budapest-by-train.htm for tips.

Looking Up Train Schedules and Routes Online gives you the Deutsche Bahn train schedule link and tips for using it, but doesn't sell tickets.

Or you can take a flight that's only 1 hour 10 min to 1 hour 25 on a budget airline like Ryanair (Google "Budapest to Prague flights").

Paul,
Five days in Budapest is like this:

Day 1 arrive by plane from the US usually between noon and four pm.
(Tired day maybe the lower end of Andrassy ut and the evening on the embankment.)
Day 2 first real sight seeing day in Budapest
(District V from Margarit Island to the Great Market Hall including Vaci utca, Falk Miska utca, etc.)
Day 3 second real sight seeing day in Budapest
(morning in Buda, afternoon the commutter train to Szentendre and late afternoon boat back to Budapest)
Day 4 third real sight seeing day in Budapest
(Jewish Pest, District VIII, Kiraly utca up to the outter loop, Liszt Academy, etc.)
Day 5 fourth real sight seeing day in Budapest
(Andrassy ut from end to end, Bath House, House of Terror, Architecture, City Park, maybe the Zoo or the Circus - or a special interest day like communism, more Jewish, WWII, museums, etc)
.
Day 6 train to Prague (takes most of the day)
Day 7 first real sight seeing day in Prague
Day 8 Second real sight seeing day in Prague
Day 9 Third real sight seeing day in Prague
Day 10 Depart for the US.

If that is the idea, pretty good. I would also look at the Czech Air flights from Budapest to Prague. Even with the check in times etc, they are somewhat faster and personally i hate long train trips.