I lived in Bavaria for four years and am very familiar with all the places you mentioned in your post. Germany has a great rail system that can get you from city to city, but then you need to move around once in a city. Munich, Augsburg, Nuremberg all have public transportation, but learning to use it will take up much time. Still, you can use it to some degree, like getting from the Munich Airport to the HBH (train station) downtown. Of course, you can walk around with a tourist map or guidebook.
I don't recommend trying to do the Berchtesgaden/Salzburg area or the Garmish/Fussen area on day trips from Munich. To do Salzburg and Berchtesgaden you need at least 3 days and about the same for Garmish/Fussen/Zugspitze.
Nuremberg is about a almost a two hour train ride from Munich, but you need more than a single day to see that city. Don't miss the
old parade grounds, site of Nuremberg trials and museums.
You can do the Zugspitze by taking the cable car or cog train from Garmish to the near top of the mountain. If you like to hike, in Summer there is a great two day hike.
In Munich you can take pubic transport to see Dachau. You can do downtown Munich on your own walking using a guidebook or map. For Salzburg, Mozart's house and the Cathedral (featured in the Sound of Music movie) and castle on the hill can be down on your own. Getting to Berchtesgaden you might look for a day tour. Not sure if the Eagles Nest is open in late September, it probably is. Late September is when the huge Oktoberfest is going on, so you will need to book lodgings early. Consider going to the fest for a day, go early when the tents are the least crowded. There is more to Munich than I have mentioned, like the old Olympic grounds and castles.
Rothenburg ob der Tauber is on the Romantic Road that runs from Wurzberg to Fussen. Several medieval walled towns are on the road as well as ancient cities like Augsburg (worth half a day for sure). You would likely need to rent a car for a couple of days.
https://www.romanticroadgermany.com
It's not too hard to see the reason for the popularity - despite the modern roots of the idea, the tour combines the historic cities of Würzburg and Augsburg with the three medieval walled towns of Rothenburg ob der Tauber, Dinkelsbühl and Nördlingen, and then finishes off with the tourist highlights of Neuschwanstein Castle and the Alps.
