Ulwell Cottage Campsite about two miles east of Swanage will give you easy access to Swanage (a lovely seaside town with sandy beaches and safe swimming), Studland beach (one of the best beaches in the country) and from there a short ferry trip straight across to Sandbanks, Poole and Bournemouth.
We live near Bournemouth and there aren't any sites right near Bournemouth. A shame as the beach is fabulous. Friends of ours stay at Wilksworth Farm in Wimborne and there is Shamba not far too. There is also a CCC site in Verwood - look really nice and a CC CL.
Loads of good sites near Swanage and loads of people seem to love South Lytchett Manor.
Good luck. We love living here, beautiful part of the country with loads to do.
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We spent 2 weeks at Wilksworth Farm a couple of years ago. Its only 7 miles from Poole and is a really nice site. The water in the pool was cold though !
Meadow bank holidays in Christchurch is a ten minute walk or two minute drive from Bournemouth (the southbourne side) ten to fifteen minute drive to town centre but longer in rush hour .The site is on UKCampsite listings not cheap but very central to Bournemouth / Christchurch