With teenagers, I would aim for Poole/Bournemouth.
Two Universities and lots of language schools mean there is lots of provision for teenagers, good transport links both within the town and to London, good schools including sixth form colleges and grammars, lots of scope for part time summer jobs, water sports and other stuff including a half-decent football team.
Culturally there is a certain amount going on in Bournemouth.. Tourists mean that a variety of shows come to the Pavilion. A new cinema complex, a reasonable pub music scene, and things nearby like the Purbeck Film Festival and Dorset Arts fortnight. The shopping is good, though scattered between out of town centres and central Bournemouth. There are huge regeneration plans for Poole Town Centre.
And the beach is lovely. Surprisingly lovely. Far far better than, say, Brighton. And the countryside around (New Forest, Purbecks) is beautiful.
There is a big choice assuming you want to be near the beach from flash Sandbanks to edgy Boscombe. But prices will be surprisingly high if you are coming from the north. Poole however is a port with light industry so away from the beach it is possible to find cheaper housing.
With a pretty open budget I would aim for Westbourne/Alum Chine. Westbourne is a bit Putney-by-Sea with a large M&S Food, good restaurants and some up-market shops. However it is well served by buses and it is a shortish walk/cycle along the prom to central Bournemouth.
Alternatively Southbourne, which is a bit cheaper but is close to the beach and leafy and has a good stock of family housing.