Salzburg recommended number of days

Depends on what you want to see. I'd want two days in the town to explore and wonder and another day to see Eagles Nest and a Salt Mine (they are SO fun) and maybe a day to see the Lake area. Vienna and Salzburg are 2 of my favorite European cities.

I hadn't done a lot or research yet can you reach the Eagles Nest and the salt mines via public transportation? Is there a specific salt mine you recommend

The easiest way to visit Eagles Nest is to take a guided tour from Salzburg. I would allow three nights for Salzburg, one day to explore the town and another day to visit Eagles Nest.
You need a minimum of four nights for Vienna and if you have an extra night a day trip to Bratislava’s old town by direct train (1h) would be ideal. If you add Slovakia make it five nights.

Yes I did think about taking a day trip from Vienna and had looked at Bratislava for a day trip.

https://community.ricksteves.com/travel-forum/trip-reports/san-diego-to-munich-just-getting-there-part-1

Perhaps my Trip Report from this spring will answer many of your questions. I know the RS format here is awkward, as the number of words are limited, so a poster must divide up their comments.
I love both Vienna and Salzburg.
Safe travels!

There should be a combo ticket at the Salzburg train station for the train, bus, and entry for the Hallein salt mine.

We bought a combo salt mine - Eagles Nest ticket on the salt mine's website (https://www.salzbergwerk.de/en) for our late May visit. Eagle's Nest generally does not open until mid to late May. We bought our bus ticket from the bus driver when we hopped on the bus in Salzburg. We caught the bus to the salt mine (has its own bus stop, but it is a little bit of a walk to the salt mine... maybe 15--20 minutes)... walked to the Berchtesgaden bus station after our salt mine visit (25 minute walk through a neighborhood and along the river)... and then hopped on the bus that took us to Berchtesgaden. Pretty easy... didn't need a guided tour to do it.

I'm outdoorsy and enjoyed walking some of the trails above Eagle's Nest but better yet, at least to me, is a visit to Königssee, a large lake near Berchtesgaden. Take the boat across the lake, walk to Obersee (an even more picturesque lake but smaller), hike around the right side of the lake to reach a snack hut, continue past the snack hut up to a really nice Alpine meadow (complete with cows), and then head across the meadow to the base of Röthbach Waterfall (the highest waterfall in Germany). Then head back the way you came back to the boat. It's a really nice way to experience nature in the area. It is also doable by public transportation.

This past Sept., I spent 5 nights in Salzburg with my sister. We had 2 full days for Salzburg and took 2 day trips. We did everything via public transportation. If you’d like to read my trip report, just click on my name. It’s the Sept. 2022: Austria and Switzerland with my sister report.

We stayed across the border in Ramsau bei Berchtesgaden, maybe a half hour from Salzburg. I’d say 2 days in Salzburg is plenty. LMK if you want more details on that area. It was magical for us.

@chris I would be interested in where you stayed and if you had access to a car. I had looked at some lodging near the border in Germany but really didnt want to rent a car to get into Salzburg and deal with parking. In addition there will be 5 of us so we would need a large car.

there are several vlogs on Youtube about getting to Hallstadt via public transport, and Rick's episodes show it as well as the salt mine

We stayed at this place. It might be one of my favorite all-time hotels because of the ridiculously beautiful, serene setting + the onsite restaurant/pub https://hotel-nutzkaser.com. We had a car so it was maybe 10 minutes to Konigsee, 10 minutes to Berchtesgaden, and maybe 30+ minutes to Salzburg. You are right about driving and parking in the altstadt of Salzburg.. it's a bit stressful w even a medium sized car. If I had it to do over, I might do one of the park & ride options on the edge of town.

Also, there is a bus that circulates around Ramsau bei Berchtesgaden, stopping at various hiking spots. From Ramsau, the bus is another 15 minutes to Berchtesgaden and another +-30 minutes to Salzburg from there.

For me it was worth it to go home to the sounds of cow bells and the smell of fresh cut hay, but everyone has their own likes and dislikes.

Adding on - we stayed on Berchtesgaden and rented our car at the Saltzburg train station (train from Munich to there to get to the area). I can’t vouch for where to leave it when you day trip in but rental and return ag the train station was really queasy and we were not on “city streets” much. I too vote that you would not need more than 2 days in Saltzburg.

My favorite place to stay in Ramsau bei Berchtesgaden is Gasthof Auzinger. It gets full very fast now that it has become more well known. They won't be open until middle May. Don't miss St. Sebastian church either. I've posted many times that I use P+R Salzburg Süd. The Euro 15 combo parking fee includes the bus fare for the day for 5 people to the city center and back. It's about a 45-minute drive from Ramsau avoiding the Austrian Autobahns.
https://www.auzinger.de/startseite
https://www.salzburg.info/de/reiseinfos/anreise-verkehr/auto/parken/salzburg-sued

Thanks for all the input. We are staying 3 full days in the Salzburg area. 2 for the city itself and then for a day trip yet to be decided.