Salzburg is a great central location to take day trips around the area. It has a great public transportation system including buses and trains as well as it is an easy city to drive in and around. It is possible to do many of the trips with public transportation although having a car can help.
While Berchtesgaden is in Germany, it is much closer to Salzburg than Munich and there is an easy bus route to get there. Bus #840 is a direct bus from the train station in Salzburg to the center of Berchtesgaden. You can buy tickets on the bus and the bus takes about 45 minutes. Berchtesgaden is a cute town that is a nice walk around with many churches and a nice downtown area. Many people go see the Nazi documentation center and Eagles Nest which you can reach using bus #838 from the train station in Berchtesgaden. Lastly and what I see as the highlight of the area, Königssee is also easy to get to by using bus #831 from the same place. Königssee is a beautiful alpine lake where you take an electric boat to the pilgrim church St. Bartholomew. This is one of the most beautiful places I have ever been and I highly recommend leaving a good amount of time to enjoy Lake Königssee. If you want more information on Berchtesgaden you can check out a previous post.
The postcard town of Hallstatt is also an easy day trip from Salzburg. The area has been mining salt since pre-historic times and now is an UNESCO World Heritage Site. I can’t think of a more beautiful village than this, and there is plenty to do as well. The best way to get to Hallstatt is by train. Not only because it is convenient, but when you arrive by train the only way to get to the town is to take a boat across the lake which makes for amazing views. Take some time to walk around the town, it is one of the best and head into the Hallstatt Ossuary also know as the bone house where human remains have been buried and skulls are painted in memory of the person. There is a small charge to go there but it is very interesting. Make sure you step into the churches as well. I enjoy getting a beer in Hallstatt, it is one of the few places you can find a Zipfer beer which is really good. They serve it at most places so be sure to try it. Many people take the funicular up to a viewing platform which is worth the money and from there you can tour the salt mines as well. Be aware, especially in the summer, there are a lot of tourist groups and buses but it is worth all the crowds.
3. A drive around the Salzkammergut region.
While Hallstatt is a part of this, I recommend doing Hallstatt on its own and doing a second trip around the region. This is best done with a car where as Hallstatt is best done with the train. You can rent a car for the day in Salzburg from the airport or train station. I would start in Mondsee which is a short distance from Salzburg. It is a beautiful town on a lake with a nice lakeside promenade. The town is very nice to walk around and the Basilica Minor St Michael church is a must see and the interior was used in the wedding scene in The Sound of Music. I would then drive to St. Gilgen on Lake Wolfgangsee. St. Gilgen is incredibly scenic and there are opportunities to get on the lake here. St. Gilgen also has ties to Mozart as his mother was born here and his sister lived here after she was married. One of my favorite spots is Bad Ischl. Unlike the other towns Bad Ischl is not on a lake but on a river. Bad Ischl was the place that Emperor Franz Josef chose to establish his summer court and met his wife Sisi here. A good was to experience imperial Austria is by going to the Kaiservilla. Lastly, I highly recommend going to the thermal baths here as well. It is a very modern large spa and is a very enjoyable experience.
While most people do a day trip to Salzburg from Munich, you can also do it the other way. Many of the highlights of Munch are very close to the train station and if you are not planning on staying in Munich, be sure to at least visit it for a day. You can read a separate post on Munich here.
Enjoy Salzburg and if you want more info on Salzburg you can find it here.