The Austrian national train company, OBB, has a subsidiary operated in
conjunction with the Czech national train company. Called “ Railjet”
and “Railjet Express,” both offer low-cost train tix on most of the
rest of your routes—
That is wildly incorrect. Railjet is just a train type and category. Where these trains cross borders they are operated by the national railways of the country they are in. So the trains are operated by CD in the Czech republic, ÖBB in Austria, and MAV in Hungary for example.
A good rule of thumb is to buy with the national railway of the country where the trip starts. That way if there is an issue with your trip you can just go to the station and sort it out with a human. Occasionally you however may be able to secure a bigger discount buying from the "other end". Eg. Luzern to Frankfurt is usually cheaper when bought on bahn,com, rather then on sbb.ch
If you want maximum flexibility then this is a trip where an EurailPass would actually be useful.
I expect you will be doing Salzburg - Zurich and Zurich - Luzern back to back (as Zurich itself is not worth including in such a trip as yours) and in that case you buy a ticket Salzburg - Luzern. This is important to remember: Tickets are for a route. You buy one ticket end-to-end in this part of Europe, even if there are mutliple trains involved.