I'm traveling to salzburg on the 14th october and will reach there in the afternoon. Planning to stay there till the 17th of October morning from where I fly to mallorca. I plan to explore around Salzburg on 14th. On 15th I plan to go to Berchtesgaden, kings lake and eagles nest by public transportation and return back to Salzburg in the evening. On 16th I plan to go to hallstatt village, dechstein ice caves and 5 fingers (or schafberg rail ride or werfen ice caves instead depending on transport availability and which are closer) again by public transportation and return back to Salzburg in the evening.

In Salzburg, please suggest on the places that can be visited.

Please let me know if traveling to these places by public transportation is a good idea and if it would be economical compared to renting a car. We are two people and had got a quote for renting a car from Salzburg airport for 120 euros for 2 days with zero excess and with a deposit of 150 euros (renting using a debit card using a company called indigo car hire based out of UK). I'm skeptical of driving around in Austria (as it would be my first time driving in Europe). I have my Indian license in English and dont have a IDP (rental company said it's not needed).

Also, please feel free to suggest any alternative places to visit if they are more beautiful and easier to access than the places I have written above or also if I should cut down on the number of places to be visited.

<<In Salzburg, please suggest on the places that can be visited.>>

The trouble is that you don't really have any time IN Salzburg itself. Even if you arrive in the early afternoon you will only have a few hours to see the highlights.

I suggest that you look at the Salzburg TI website to start off with and work out what is feasible within your timescale. You should also be able to find public transport links and also links to commercially arranged tours for your two excursion say which may be a better bet considering your tight time schedules and lack of experience travelling in Europe. That's no criticism BTW - just that sometimes tour companies can be a better option.

https://www.salzburg.info/en

"In Salzburg, please suggest on the places that can be visited"
Why?
You didn't plan any time for the visit of Salzburg anyway.

Is there any way you can add some time for Salzburg itself? Arriving in an afternoon and only having the rest of that day for it is really short. It is a beautiful city, so if you could even squeeze it another half day that would be better IMO. Be sure to take the funicular up to the fortress. Also the Mirabell gardens and palace. Our favorite thing though was literally just wandering the city.