Should I rent a car in Austria

If you only need transportation between cities, then the train or bus will work for you. I use two web sites to see how to get around: rome2rio.com and bahn.de. The former looks at trains, buses, cars, planes, etc. while the latter has train schedules. Having a car in most European cities is a hassle as there is limited parking.

I agree. Our car sat parked while in Salzburg. You are able to get around on foot or with the local transport. In Hallstatt your vehicle remains outside of the town center in a parking lot or garage. You either walk into the town or take the hotel shuttle. You can easily take a train between Salzburg, Hallstatt & Vienna if those are your main stops.
Enjoy your trip.

Being a major ski resort, Kitzbühel has frequent rail connections to the rest of the country. The town is manageable on foot, and at least one gondola sits a short walk from both the Bahnhof and Altstadt. Your rental car would sit in a garage the entire time.

No car needed. Austria has a terrific public transport system. Bahn.de (German train) is easier to find schedules but if you want to buy advance tickets go to OEBB (Austrian rail).

I've spent five full or partial days traveling in Austria, all by train or bus and got to everywhere (Salzburg, Mondsee, Hallstatt, Hall, Innsbruck, Kufstein) I wanted to go. When I looked at your list of places (Vienna, Salzburg, Hallstatt, and Kitsbühel) you want to go to, my first thought is that there are trains to everywhere on your list.

The Bahn website does not show connections in Austria by bus, but the OeBB site does. Before committing to a car, make sure there is someplace you want to go that you cannot get to by public transportation.

In 2009, I went from Salzburg to Hallstatt, but I wanted to see the SOM wedding church in Mondsee, so I took a bus from Salzburg to Mondsee, then went by bus to Bad Ischl via Wolfgangsee. I could have taken the train from there to Hallstatt Bhf, but since my accommodations were in Lahn, 1 km south of Hallstatt, and the bus terminal is in Lahn, I continued on from Bad Ischl to Lahn by bus.

Thank you so much all for the replies and the information...much appreciated..