Advice for base town for hiking in Austria without a car

Seefeld in Tirol has good hotels and plenty of walking and hiking of all levels, with a funicular starting the trip up to altitude. Very popular with Germans, and among others, me.

It is directly on the Innsbruck - Munich train line so very easy access to Garmisch and others.

You might look at the Werfen area. I know they offer paragliding in the area where you take a tram up to the starting point. There is an ice cave in the area. St. Johann im Pongau is a ski area close by with lifts. (This is a different St. Johann than previously mentioned.) We stayed a week at St. Johann im Pongau with a car and day tripped all over: Salzburg, Hallstatt, Berchtesgaden. Probably just as easy by train.

Though I don’t think you will find the small villages nestled into the mountains like Wengen. The towns are mostly in the valleys, surrounded by rolling hills and the mountains a little farther out. Not like Lauterbrunnen valley where the mountains ascend so steeply. You may find transportation a little more challenging.

Nigel mentioned Garmisch - good suggestion. Its in Germany, but definitely worth looking into. Check out Mittenwald, super cute town with mountains towering above.

"Seefeld in Tirol" is the hiker's paradise, offering 200+ hiking, walking trails. About 100 can be found here:
https://www.seefeld.com/en/summer-3/walking/hiking-trails

Seefeld has train connections to Innsbruck, Mittenwald, Garmisch-Partenkirchen, and Munich.

We spent a week in Fieberbrunn in the PillerseeTal region of Austria for a winter event. We arrived in Fieberbrunn via train from Salzburg and enjoyed free bus service connecting the five village region. We were able to take a free bus from our hotel to one village, ski to another and return to our hotel by free bus service.

Hiking: https://www.kitzbueheler-alpen.com/en/pital/so/hiking/holidays.html

Free local bus service: https://www.kitzbueheler-alpen.com/en/pital/mobil.html

With your valid guest card, the buses in PillerseeTal as well as to
Kitzbühel, Ellmau and Lofer are included. The local trains (except IC,
EC and RailJet) between Hochfilzen and Wörgl (Kirchbichl) can also be
used for free. Simply show your valid guest card to the bus driver or
conductor and explore the region at no cost!

There is also the Zillertal. Long broad valley with mountains either side and a train line down the middle, Zillertalbahn. I've never visited but it is definitely on my list. Famous for German tourists and plenty of local music.

2 more options, both about 1,5 hours by train away from Salzburg.

Zell am See
As the name says it's located near a lake.
The local mountain is called Schmittenhoehe https://www.schmitten.at/en with several walking and other activity options.
If you want to go up higher into the snow Kitzsteinhorn https://www.kitzsteinhorn.at/en/explore/summer/fun-action would be the place to go.
Day trips by bus to Grossglockner high alpine road possible. https://www.grossglockner.at/gg/en/index

Bad Gastein
on the main train line Salzburg to Venice. You can directly change from train station to gondola to Schmitten Höhe.
https://www.schmitten.at/en
It's not located at a lake but has a Thermal bath in town. It has the charme of old Habsburg Monarchie but some of the old buildings are rotten.
https://www.gastein.com/en/gastein/gastein-valley/bad-gastein/
Gastein valley is full of walking options.

Quick note:

The Austrian Alps aren't the Swiss Alps. So you won't find a near analogue of the Berner Oberland. But the Austrian Alps are nevertheless incredibly special and beautiful. Maybe not the crazy glaciated craggy high alpine of Switzerland, but so green it burns your eyes and so sweet and charming and cozy you weep into your kaiserschmarrm. I think I like looking at Switzerland more, but I prefer hiking in and simply enjoying Austria.