It depends what you want to do most of the time really. Shops, restaurants, good transport links but not great beaches - Nice. However Nice is also very touristy, very hit and miss for restaurants, and due to the huge number of visitors not very "French' in the summer if that is what you are after. Cannes isn't too expensive depending on where you stay. The big hotels are extortionate, not so much the small ones and apartments for rent. You also have the Ile des Lerins, great for a day trip and other things going on in the summer. It is a good mix of a very French town (more 'French') than Nice, and a bit of luxury.
It also depends,on the time of year, if you can go outside French school holidays it is much cheaper.
Antibes is nice but packed in summer.
Do you want somewhere quiet or more lively?
Regarding the pool, only you would know how much you would use it. If you are going in the height of summer most pools are only good to cool off in, so full of kids you can't swim!
I would look at some of the smaller places. St Raphael is very nice but a bit far if you want to visit Nice and places to the east, it is also very small if you are wanting somewhere a bit lively. Villefranche is very nice, and Menton, close to Italy is lovely, but again a bit far east if you want to visit towns such as Cannes.
Mandelieu La Napoule is very close to Cannes, with nice sandy beaches and a few nice restaurants. It is busy in summer but not so much as other places, relaxing even in the main holiday weeks. I love it just as well seeing as I live there!!