EDIT: Okay we crossed paths with out posts. Still not sure when you land in Vienna or depart Vienna. I have an extra day for some reason.
Okay, so dates do help with the suggestions. I presume you have a round trip out of Vienna? That might have some impact. Then just so I understand your 12 days are like this:
1 Leave the US (or was this not in the 12 days?)
2 arrive Vienna
3 – 11 in Europe
12 Depart Vienna
Its fine for what you want to do but it is a bit tight which is why every day gets counted correctly.
If you are coming in November I have some suggestions, if you are coming in August I have some different suggestions, so the time of year matters. Then, for Austria, Emily might know some special events or one-off happenings in Austria, so that’s why the exact dates are useful.
A quick basic run-through might be what is below or it’s a place to start at least. This gives you 4 full days in Vienna and 2 full days in Salzburg, and I think thats good. You get very little time in Hallstatt, but ……. Maybe give up one night in Vienna for Hallstatt. To be honest, I was glad I went to Hallstatt last September, but since then they have closed the mine and the skywalk and the funicular up the hill. So that takes a lot out of it for me. I also went to Salzburg last September and I will confess a moral crime …. I didn’t enjoy Salzburg. Glad I saw it I guess but would never go back. A real disappointment as I was looking forward to it, but not so much Hallstatt that I ended up loving. Strange how that works.
September 27 Arrive Vienna (Night in Vienna)
28 Vienna (Night in Vienna)
29 Vienna (Night in Vienna)
30 Vienna to Salzburg (train of course) (Night in Salzburg)
October 1 Salzburg (Night in Salzburg)
2 Salzburg (Night in Salzburg)
3 Train or Daytrip.com to Hallstatt. I like the Daytrip.com because you can make stops and make a real sightseeing trip out of it. But you will arrive later that way. Spend the night in Hallstatt (night in Hallstatt)
4 Hallstatt to Vienna by train (night in Vienna)
5 Vienna (night in Vienna)
6 Vienna (night in Vienna)
7 Return Home. (thats 12 days)
September 8 ??
Your idea of a day trip to Budapest has me torn. If you do it you will become pretty angry with yourself that you wasted time in Salzburg (maybe, maybe not). It will be a hard day. I am going to suggest that if you do want to do it, then you get an early train, maybe 8:00 am out of Vienna. That will get you here before lunch. Between myself and a few others here we can map out a day that at least will show you why you should have stayed longer and convince you to come back. BUT THEN, do not return. Return for what? To go to bed? No, get a room in Budapest down on the river with a view. I am assuming warm weather so sit outside on the Corso for dinner and live gypsy music and discuss your regrets of not having stayed longer. Then the next morning catch a 8:00 am train back to Vienna. Or if you are having a great time, you catch a later train. Trains are cheap. You do this, you miss a day out of Vienna. If I guessed wrong about Day 1 being a travel day and Day 1 is a day in Europe, then maybe it works. Or cut a day out of Salzburg.
Prague. You don’t have the time for the journey.
Favorite Tourism Video of Budapest https://youtu.be/1nd5AtZIrTk?feature=shared Anthony Bourdain describing how visually pleasurable Budapest is https://youtu.be/0nd9DuDGCz0?si=aa0c19KaJHEtabkX