When, Where and How to buy tickets in Eastern Europe

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3 days in Vienna, 4 days in Budapest, via Bratislava, 3 days in Krakow (via night train), 3 days in Prague (via night train)

I don't see any way to buy good priced tickets from the USA, is this correct? If yes, then must we wait until we arrive in each city to buy the tickets out of the city? This makes me nervous travelling with children and on a strict schedule. Or can we buy the tickets for each leg of the trip at once in Vienna? Are there international ticket windows in all major city train stations?

One last question; Can you see any cost or convenience advantage in reversing the itinerary above?

I don't see any way to buy good priced tickets from the USA, is this correct? If yes, then must we wait until we arrive in each city to buy the tickets out of the city?

Yes. The only part of your journey where you can use tickets purchased online and in advance is Vienna - Budapest (e-tickets from OEBB website), for other legs you don't have any possibilities to do it besides buying in Budapest, Bratislava, Prague etc.

This makes me nervous travelling with children and on a strict schedule. Or can we buy the tickets for each leg of the trip at once in Vienna?

1) I wouldn't worry about it at all since you spend 3-4 days in every city before departing to the next destination, so just purchase tickets first thing after your arrival - buying 3-4 days in advance is definitely enough time. Plus in case something extraordinary happens you always got other options - like take evening train instead of overnight etc.
2) You can purchase all the tickets in Vienna, but the price of those will be really high since they will be issued with a help of standard TCV tariff, while purchasing in Hungary, Slovakia, Czech republic gives you chance to use special interstate discounts (not available if you do it in Vienna)

Are there international ticket windows in all major city train stations?

Yes, all the main train stations will have international ticket window, but in some cases (if the station is not the main, like Budapest Nyugati or Deli) international ticket counters will be open only during day time (like 8am-20pm or so)

One last question; Can you see any cost or convenience advantage in reversing the itinerary above?

In case of reverse route (as far as I understand you mean Prague-Krakow-Bratislava-Budapest-Wien) the difference isn't that crucial. Mainly - you can get 13 euro ticket for Budapest-Vienna route instead of 19 euro Vienna-Budapest, save something on Krakow-Bratislava route with prepurchased Czech ticket Bohumin-Bratislava etc. But, once again, all these aren't that essential things.
If you ask how I would do similar trip I would say probably Prague-Vienna(daytime due to cheaper 19 euro tickets) -Budapest-Bratislava-Krakow(overnight) or Prague-Budapest(daytime due to cheaper 19 euro tickets)-Vienna-Bratislava-Krakow(overnight) - the idea behind is to start/finish with the farthest destianations of your itinerary (in your case those are Prague and Krakow). But in your case (since you are from States) it all depends on your starting/final points.