No grammar system. The secondary schools (state) are very different in character but you generally hear good things about Ralph Allen and Hayesfield, St Gregory's (Catholic) too. Beechen Cliff has been in the news with some serious issues you would definitely want to research.
For primaries then Bathwick, Widcombe and WASPS are generally considered very good (but then their intake demographic makes that likely). Also heard good things about St Stephens, and I hear St Andrews, which has a very different demographic, is up and coming.
Private - KES Prep is excellent as is KES senior (academic). Royal High School also an excellent academic reputation.
There's also other private schools, all with their own character, such as Prior Park (Catholic), Monkton (Anglican), Kingswood (sporty).
Pollution - fairly high around the main roads through the city, and it's in a bowl with hills around so can trap. May improve with recently introduced clean air zone (which, beware, may one day include private vehicles as well as commercial which will make driving within the city expensive).
Traffic - Can be a nightmare, such as 45 mins+ to cross at rush hour. Has improved to some extent in recent years. Obviously quieter than normal at moment.
Parking - Dreadful. Very small zones, requiring permits, and even then you're fighting to get a space and may find yourself parked several streets away.
Areas -
Widcome and Bathwick are nice (and expensive), and very close to the train station so popular with ex-Londoners.
Bear Flat (Edwardian/Victorian) is popular with middle class professional parents, also fairly expensive.
Georgian parts of Lansdown and centre of city are very expensive, and either small flats more suited to young professionals, or £2-5m townhouses.
Camden is nice (Victorian terraces mainly) but quite a walk from the city/train station.
Larkhall (Georgian + Edwardian and 1940s) also quite far out but nice houses and villagey feel. Again quite expensive.
Weston is close to the hospital, far from the train station but often more spacious.
Cheaper areas - Twerton (can be a bit dodgy, but only in Bath terms), Oldfield Park used to be very studenty but many students are moving to purpose built accommodation so I think it will go up in price and atmosphere. Fairfield Park.Odd Down. All these are quite a trek from the city centre.
Public transport is pretty poor. There's a train line that runs from Bath Spa to the small Oldfield Park station on the southwestern edges. Otherwise it's buses and except for the centre-hospital route and university route, they're infrequent and unreliable IME.
Good links to Bristol, Bristol airport, London etc.
As it's a hilly city (aside from very centre) e-bikes are very popular!
To live in it feels like a lovely, safe city, though a little insular and old fashioned at times. Very popular with ex-Londoners escaping for a more relaxed pace of life (around 90% of my DC's classes are 'incomers' either from London or other university cities, though will depend on school). It's very liberal, well-educated, open-minded. Unfortunately it's not particularly multicultural (it's overwhelmingly white) but the university does bring some diversity.
Despite the lovely Theatre Royal, there's less culture than London, Bristol, Edinburgh, Manchester, so for most shows/opera/ballet/comedy/rock, you need to travel. Trains back from Bristol stop surprisingly early and can be a rush to make it back from a show.