Because the Austrian national train company does not appear to offer tix for the entire Salzburg to Budapest route, you might buy these at www.TheTrainline.com
This is not correct. There is a direct RailJet train every two hours from Salzburg to Budapest. Of course, OeBB (Austrain Railways) offer tickets for the trains they operate.
Buying tickets from resellers, like The Trainline, is discouraged, as already often noted here.
Anyway, I would switch Salzburg and Vienna, i.e., Prague-Salzburg-Vienna-Budapest, to minimize going back and forth. Using this sequence of cities there is only one small part (between Linz and Salzburg) you will travel twice, because the fastest train connection between Prague and Salzburg (5:28 hours) reqires a change in Linz.