I'm not super-local, but often ride through the New Forest to Bournemouth and have days out on the beach, so I guess I have a visitor's perspective. When Bolletta was tiny it was a favourite destination, so I've got some happy memories.
The good thing about Bournemouth is that it's effectively a small city by the beach, so you have all the amenities.and the transport is also good, so you can get out and about if you need to. Unfortunately it seems to have been hard by the recession so, although the public spaces are well kept, the high street areas are looking very, very shabby. It can also get a little lively in the evenings. The Odeon in the centre is a flea pit.
The opportunities for family riding are good - the ride along the beach path from Christchurch to Poole is doable for littlish legs (about 4m climbing total) but still feels like an achievement.
Eid Mubarak for tomorrow BTW (or Friday - make your mind up! ;-) )