I heard that you can buy 1 train ticket from Budapest to Munich and stop at cities along the way for a day and then continue on, all on the same train ticket. Does anyone know how to book this?
I would buy the ticket at the train station or a travel agency in Budapest. There would be no advance-purchase discount, since any offers of that type require locking in travel dates and times. Trains on that route shouldn't need reservations. However, different speeds of train can have different prices, so you'll need to know whether or not your open ticket can be used on RailJet service or only departures labeled as InterCity or EuroCity. Cost savings to buy this as one ticket versus three separate tickets is probably minimal. Travel must be completed within a window of two weeks.
You can book it at www.bahn.com. Problem is it really gives you only 3 hours max at each stopover, barely enough time to get any kind of a feel for the cities and take a couple of pictures. It is a 7 hour journey without stopovers. Add 6 plus hours for stopovers, and it is about a 14 hour day.
But it can be done for as little as 39 euro if you book it 90 days in advance.