Palma itself is a nice city, but there are not many places for a family beach-type holiday in the town itself. You'll probably be looking at a seaside resort town.
There are very few heated (outdoor) pools in Majorca, and they tend to only run the heating in May and October. But if you're thinking of going in the next 3 months, don't worry, the pool will be warm enough. In fact some people complain in July and August that the pool is too warm to be refreshing. Most larger hotels have a kids' pool, but you would want to check with the photos on their web site.
Also don't worry too much about distance from the airport if you find a nice-looking place. The major thing determining how long a transfer takes is luck, if you get a coach transfer. It can take between 0 and 90 minutes between you getting on the coach and it leaving the airport, depending on the order of flight arrivals (they fill up the coach from several flights). And it can take between 0 and 60 minutes to tour the hotels when you get to the resort, again depending on luck. Compared to that, the actual driving time of between 30 and 75 minutes is probably not the main factor. If you want to avoid the coach lottery then get a taxi, or if by some chance your hotel is near a stop where the airport bus goes, get that.
With that said, if you go to somewhere not too far from the airport, the taxi option becomes more affordable. To Playa de Palma is only €20 and there are some nice hotels, but a majority of the visitors are not English-speaking (although that might or might not be what you want). To Palma Nova/Magaluf/Santa Ponsa is €40-50 and those places have a lot of Brits. (Don't worry about Magaluf, it's not Saint Tropez but it's not Gomorrah either.)
If your kids really like the water you could consider the newly-opened Fergus Club Mallorca Waterpark in Magaluf (ignore the reviews from 2023 and earlier when it was a stag-do hotel), but I just looked at the prices and it's, umm, not cheap. There are other hotels with built-in waterparks too (Katmandu Park in Magaluf, Splash World and La Reserva in Sa Coma).
If your budget is a bit tight then look at the all-inclusive places in Calas de Mallorca. This is a purpose-built resort town about an hour's drive from the airport with not much charm, but the AI hotels are good value and there are a couple of decent beaches within walking distance.