We are staying in Salzburg in early August. We plan to take the bus to the Eagle's Nest and do a tour there. After the tour, we would like to spend some time in Berchtesgaden, have lunch and tour a salt mine before returning to Salzburg. Will we be able to get public transport to do all of that? We are open to trains/busses/taxi or Uber. Trying to avoid renting a car if possible.
In Aug 2025 we did a day trip via public bus from Salzburg to Berchtesgaden to join a Public Eagles Nest Historical Tour visiting the Eagle's Nest and some associated sites; the tour started at 1pm. Before the tour we spent 90 minutes at the worthwhile Dokumentationszentrum Obersalzburg and had a very quick lunch. This took almost all day, as we left Salzburg on a 9:18am bus and arrived back shortly before 6:30pm. Though I don't know the length of your tour nor the location of the salt mine, the schedule seems a bit tight, at least via public transport.
More details of our schedule on that day are in our trip report (part 6): https://community.ricksteves.com/travel-forum/trip-reports/trip-report-austria-bavaria-aug-sep-2025
We took the public bus to the salt mines, on the way to Berchtesgaden. Do get tickets in advance, as times sell out. Then the public bus continues to the lake, and beyond. Also, take the boat trip on the lake, and hike to the falls-absolute beauty!