I have name changed especially for this, as though I VERY seldom post, my love of Salzburg is well-known!
You don’t mention your budget, but re a hotel the first thing I’d say is that public transport is excellent, so any hotel in the centre will be accessible. I like staying in the ‘New town’ - and there are lots of great options. Anything on/off Linzergasse, Schwatstrasse or Mirabellpatz will be great. Master serviced apartments have 2 separate buildings in this area and their apartments are hotel standard with more living space and a little kitchen area - I’m happy to eat out when travelling alone, but it’s also good to have the option to eat in.
Re activities- the SOM tour is actually a great way to see the countryside, and there is a newly opened SOM museum too. Apart from that the big thing is Mozart - either his Wohnhaus or Geburthaus. I really like the Freilichtsmuseum which is a bus ride away - a living village type museum, I’ve been there in Winter and it was still lovely.
Other places/things - Mirabellplatz, the Festung (fortress) walk up the Kapuzinerberg or Mönschberg. Wander Getteidegasse, Goldgasse and definitely have Kaffee und Kuchen - Tomaselli is the famous one, but I have a nostalgic affection for Schatz.
Transport in the whole of Salzburg province will be free with your hotel, so get on a bus and explore - St Gilgen or even as far as Bad Ischl are lovely and the bus will be free!