They're quite different geographically!
Leamington is the smallest of the 3, but probably the best connected. You can be in Birmingham in 45 mins, London in 90 mins, Manchester in not much more, Bristol in 2hrs... but it's a long way to the sea.
It's a nice town, with plenty of choice of good restaurants, pubs, cafes etc. Shopping isn't as good as it was - but that's probably the case everywhere. Plenty of out of town shopping for the big stores, and a decent number of independent retailers - but it's a tough time to be on the high street, and there are notable gaps.
There are nice green spaces/parks, a decent cinema and a small theatre (but you've got the RSC up the road, so you're not short on culture). Warwick is just down the road, and you've got Stratford 20 mins away.
The housing stock is very varied - everything from huge Regency mansions, to plenty of mid-war kind of stuff, to the new build estates of the 70s and 80s and a HUGE amount of brand new stuff springing up.
I can't comment on the schools specifically (friends kids have all gone through the school system now. Secondary is a long way off for you, but North Leamington School has served friends' kids well, as has Myton (which is in Warwick, but depending on where you live in Leamington you might be in catchment).
If you want a town, rather than a city, it has a lot going for it. We moved here kind of by accident 26 years ago, and have been very happy here.