Hello - mom here as well as long time Austria resident! Based on your interests, I would say that you want to spend time outside of Vienna (which I think is a good idea as there isn't much here that appeals to kids, in my opinion).
I'd go beyond the obvious suggestions mentioned above (Vienna, Salzburg and Innsbruck), especially as these are cities.
I'd choose to base in 1) St. Gilgen, 2) an Alpine town (Ischgl, Soelden, Mayrhofen, Serfaus, Seefeld) and either the Wachau or Loipersdorf.
Upon arrival, I highly recommend that you DO NOT get in a car and drive as that would be very unsafe. Spend the first night in Vienna and then head out to your chosen destination. The same with the return - plan to spend the last night in Vienna. There are loads of car rental agencies with offices in Vienna, so you do not need to rent the car at the airport. As a general comment, while public transportation is insanely good in Austria (heck, I haven't driven a car in 12 years), you can see a lot more in rural areas if you have a car. Do not try to use the car in Vienna except to get out of town.
St. Gilgen - My favorite town on Wolfgangsee, with great restaurants, a swimming area and a good crafts market, if you are lucky. Cablecar right up to a summit is in the town for hiking. Take a short bus ride for a day in Salzburg (mandatory), visit Hallstatt, ice caves, St. Wolfgang, rodelbahns. I'd highly recommend staying in a farm in this area - look at this website - https://www.farmholidays.com/
Alpine Town (suggestions of towns listed above) - All of these towns are quaint, with excellent views and great hiking/summer sport infrastructure. These towns have loads of AirBnBs as well as resorts.
I was torn here between the Wachau (the Danube River between Melk and Krems) and the area around Loipersdorf (a spa area with loads of charm and attractions). I think with a kid, I'd lean towards Loipersdorf. This is a super fun area for swimming with a huge complex, but especially great for the apple, pumpkin and wine harvests (you'll be just in time). The best attraction for kids in all of Austria is also nearby - Zotter Chocolate Factory. Like Willy Wonka's - no lie. Riegersburg Castle is stunning and not to be missed (the birds of prey show is great for kids). For the parents, this area is known as Austrian Tuscany, with rolling hills, vineyards and apple orchards. Nice biking here too.
So that's my pick for 3 bases. You could connect these three locations for a nice loop.