Greg,
If I were taking this trip, I'd travel via rail on the routes Zurich-Salzburg, Salzburg-Vienna and Vienna-Prague. The longest trip is only about four hours or so. However, on the Prague-Zurich trip, budget air would be a better option unless you have lots of time (the rail trip is about 11 hours!). I believe GermanWings flies that route, and some of the fares I saw (which were listed in Czk$) were only about €125 for a one way trip.
I never bother pre-buying rail tickets (except of course for Rail passes). It's very easy to buy P-P tickets when you arrive, at the first large station you encounter. I prefer that method as conditions can change on the rail networks between the time you buy the tickets and the time you'll be travelling (I experienced an example of that this spring on a trip from Salzburg to Hallstatt - some of the rail lines were "under repair").
In your circumstances, this is the method I'd use:
After going through Passport control in Zurich, buy a one-way ticket from Zurich Flughafen to Salzburg. Depending on your time of arrival and which train you choose, there will be at least one change at Zurich Hbf. The trip to Salzburg will be about six hours. You'll be tired after the flight, so that's not a time to worry about tickets and other details. After you arrive in Salzburg and get settled and more importantly well rested, take a quick trip to the OBB office at Salzburg Hbf a few days before your outgoing journey, and buy tickets for the two remaining rail journeys.
For the air trips, pre-buying is the best option as most budget airlines operate on a "sliding scale". As the seats sell, the price increases. I've never flown with that airline, but hopefully someone else here can provide some user comment. Packing LIGHT would probably be prudent!
Good luck and happy travels!