Hello and welcome to Geneva!
Your question is not easy to answer
So first of all, if you want to send your kids to the Swiss public schools (most of people, expats included, do so) you get the school the government gives you based on where you live. If you want to send them to private/international schools you can see on google where the main ones (Florimont, ECOLINT, etc) are.
In terms of where to live: the north side of the city (everything on the right bank of the river/lake A.K.A. the Airport side) is quite tricky. Now with COVID and less airplanes some places might seem very nice and quiet (like Bellevue) but when the airport will be back to full capacity you will regret it. Essentially look at the map and anything between Versoix and Satigny and going south until you get to Petit/Grand Sacconnex is quite airplane-noise bad. So avoid Vernier, Meyrin, etc...
The second thing to consider: where will you work? Living on the south bank of the lake and work in the north bank means a lot of traffic to consider. Again, with covid now things are better but traffic is quite intense during rush hour.
Regarding house prices, Geneva is very expensive. In the West part of the canton there is a countryside area with many villages, but please consider that if you live there you'll need to take the car to just to go to have a cup of coffee.
Less expensive "suburbs" are the villages in Vaud between Geneva and Nyon or even a bit further east Nyon.
Main issue is that then you need to commute from those villages to your place of work in Geneva. Traffic on the road there is the worst (it's the highway with most traffic in Europe IIRC) and if you take the train you need to consider the cost of the train ticket and then if your place of work is close to a train station or you need to change to a bus or even multiple buses.
With all that said, I would suggest you to look in the Petit Lancy, Onex, Plain les Outes, Cressy, Bernex, Confignon... Quite suburb, not the prettiest, lots of schools, close to Florimont international school and the Lancy international school, lots of public transport, no airplanes on your head.
In any case remember that in Geneva the housing market is quite crazy so you don't always have a lot of choice where to live.