Hi there
I come from Bath although haven't lived there now for over 14 years but go back regularly to visit.
It is a beautiful place to live with amazing restaurants, shops and clubs.
It can be a bit snobby and has a higher cost of living than Bristol.
In terms of areas, there are some that are not very pleasant as there is in every city.
Lansdown area has some beautiful properties and is very upmarket and nice although you are on the outskirts of the city. Any of the surrounding villages up there are pretty as well but very rural so may not be your thing.
Newbridge has some nice bits but as you said you will always be near some traffic.
Combedown is very, very nice and is situated near a number of secondary schools in Bath. Larkhall is supposed to be pleasant but I don't honestly know that area. Anywhere near Claverton Down is also nice and around Combedown.
There are many beautiful properties along near the Victoria Park particularly along Weston Lane and near the golfcourse. This has the added bonus of being near the city centre and near the Royal Crescent.
Weston is a nice village area which has some pretty upmarket areas but also some not so pretty but is reasonable and has good transport links. In fact there are good transport links all over the city should you need to use public transport.
Areas which don't have the greatest reputation and are not populated by period properties are Twerton, Foxhill, Snow Hill, OddDown. These areas tend to have 'social problems' and are not as pleasant. Walcot is the 'artisan quarter' and therefore has a mixed socio-economic population and is the 'trendy' area.
Erm... as for schools - not sure but I know when I grew up in Bath, the secondary school choice was quite limited. There was only one non-denominational mixed comprehensive which was heavily over-suscribed and difficult to get into because of this, the rest were either single-sex, catholic or private which may not be a problem for you but I know was a major issue for the population.