As with any of the train connections, make sure you don't take the slow train connections that stop on every village. There are fast (and really fast) trains, but local trains may take quite some time to cover a distance.
Another way to stop in Salzburg is to go to Munich first (see above, direct connection from Lucerne to Munich), and after your visit in Munich take a regular train to Salzburg. You stay in Salzburg for the day, and then board the same night train to Budapest that originates from Munich in Salzburg instead. The night train arrives in Salzburg around 1:20AM, but doesn't leave until 3:30AM. You're still allowed onto the train when the train arrives, this way you can board your cabin and sleep, while the train is being recombined.
=== Next stop ===
Munich and the surrounding area is next.