Budapest,Vienna and Salzburg in 7 days

Arrive Budapest, 4 nights in Budapest.
2.5 hour morning train to Vienna. 2 nights in Vienna.
2.5 hour morning train to Salzburg. 1 night in Salzburg.
Fly Home from Salzburg

This way you are staying half to 2/3rd of what each stop usually gets.

(SKIP Salzburg or Vienna)

Do you already have your airline tickets?

First, it's important to know where you are flying into and out of. To save time, most recommendations, such as Mr. E's excellent post, will focus on on traveling east to west or west to east without backtracking.

Second, you have to decide what is important to you in order to set the number of days per city. Budapest and Vienna are both great cities, but are different expereinces. For example, I enjoy Impressionist art. Vienna has a great museum for this, Budapest does not.

Budapest and Vienna are both large, great European cities, and they're worthy of 4 days or longer each.

It all depends on where you're flying out of. People in Salzburg that are flying back to North America usually go through the Munich Airport. Just getting to Budapest Airport is not the easiest.

I would suggest visiting Budapest and taking a train up to Bratislava. Then take a train over to Salzburg and flying home from Munich Airport (2 hr. train ride from Salzburg.)

In reality, you only have 6 full days, (Sun-Fri.) And even at that, it is less because of checking in and out of hotels and travel times. Hopefully you have an open jaw ticket. I agree with Mr. E. that you need to drop either Vienna or Salzburg, as you do not have enough time for those three cities. I would suggest you do more research and consolidate your trip to a maximum of two locations.

Don't get me wrong, no criticism of the pace. Not my trip. Just trying to help make it work. The OP mah love it.

Now if it were me, I would do one night in Vienna, then off to Sslzburg. But my taste is peculiar.