In the order you're doing it, the Vienna-Prague-Berlin bits would be best (more cheaply and quickly) done by bus or train than by flying. For trains, see europetrainsguide.com/Advice/How-to-Buy-Trai… for Vienna-Prague and https://www.cd.cz/eshop/default.aspx (switch to English in the top right corner) for Prague-Berlin. both fares are 29 euros. Buses from Vienna to Prague can be cheaper, such as http://westbus.at/de.
The Berlin-Budapest bit would be quickest by plane (try Easyjet: http://www.easyjet.com/en/). There's also a night train from Berlin to Budapest which would save you a night of accommodation, book that at http://www.bahn.de/p/view/index.shtml
I think hiring a car and driving would be very expensive and not worthwhile, especially if you plan to pick up and return the car in different countries.Edited: 07 August 2014, 15:35
use the suggested links above....
only buy train tickets via respective railway companies...do not buy EuroRail pass
How far in advance should we get our train tickets for the train from Prague to Berlin?
On Czech railways site you can book from 60 days in advance, on German from 92 days...but check prices, CD sells special offer cheaper.(mostly)
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