Tunbridge Wells is a lovely place to live and bring up children. There are loads of parks, woods and the common for days outside. Lots of children's groups and great countryside. The only downside is the school catchment areas for primary and the 11 plus for secondary.
The village area around the Grove Park is very pretty, close to high street and station. Downside is the catchment for Claremont has shrunk in recent years but that may change in the future. Also parking is difficult if that matters to you.
St James is lovely with parks nearby and a good school.
St Johns is also a nice area - great park with a sandpit and a well regarded primary school with three form entry. Walkable into the centre of town. It is cheaper than the village area. It is also well placed for the Grammar schools.
Bishopsdown area is nice but Bishops Down Primary only took siblings last year (2014).
Roads off Mount Ephraim - Molyneux Park, Court Road, Earls Road are nice but £££ and close to Rose Hill if you are thinking of going private.
There is a free school which opened a couple of years ago on the old hospital site and they are building lots of new houses there.
Southborough has a good primary schools and is a bit cheaper than other parts of Tunbridge Wells.
Villages with good primaries are Langton Green, Pembury, Groombridge and Frant - all within easy reach of Tunbridge Wells.
Good luck with your search