Isle of Wight - what area to stay?

Shanklin is good for swimming with generally flat water and a sandy bottom. There arepedalos and similar floating things to hire, as well as an array of booths selling sugar in various shapes, as well as chips served with extra chips. Shanklin's esplanade has a mini selection of funfair type amusements plus a large crazygolf. There are toilets at both ends of the beach (the ones nearer Sandown Pier are better), plus a big car park right by the seafront. Around Shanklin you will find places to hire kayaks for a bit of holiday larking.
The Luccombe Chine end of Shanklin Beach (only the section starting from the bottom of Shanklin Chine headingright as you look out to sea) is dog friendly all year - a big sandy area at low tide, toilets and parking at the Esplanade.

I would second Shanklin. You can stay right on the seafront or up on the cliff. More going on than Bembridge plus you can walk to Sandown along the beach. Cowes has some nice shops but the beach isn't much to write home about.

I think it depends which dates. If it's around Cowes week then near Cowes would be good (although accommodation high cost) as there's a lot for teenagers to see/do.

Otherwise, I like Ventnor if you're staying in the town as teenagers would have a lot of freedom. Ventnor has a sort of festival (carnival) with shows/music etc which means the town has a bit of a buzz. However, after a few days they may want to explore further afield. Bus service on the island is generally good though.