Many of the traditional winter mountain resorts in the Tirol and Salzburgerland now also have extensive summer programmes - hiking, biking, mountain biking, zip wires, outdoor pools, Go Ape-style climbing parks, summer toboggans (on rails), paragliding, rafting etc. Some resorts have special "Tourist Cards" for visitors that include reduced prices (or even free access) on buses, cable cars, pools, museums.
The tourist information offices have good websites that usually also include an accommodation booking service, or you can look at Airbnb, Booking.com, vrbo etc. for self-catering. Austria has lots of basic but clean self-catering apartments, and "apartment hotels" which might also have a restaurant/pool/spa etc.
Flying onto Salzburg puts you within 60-90 mins. of plenty of resorts - e.g. Saalbach-Hinterglemm, Zell am See, Kitzbuhel in the mountains to the west, or the Wolfgangsee / Salzkammergut lake district to the east of Salzburg.
Vienna is on the other side of Austria from the mountains - 3-4 hours by car or train from Salzburg. Vienna is well worth visiting, but the travel will take a day out of your holiday and you'll need at least 2 days there. There is plenty to see in Salzburg if you want to spend a couple of days in a larger town.