St Helier, Jersey, Central Accommodation.

DevCoder

I live in Jersey and can recommend a few places

The merton, very nice, has swimming pool and other facilities, on outskirts of town probably 2-3 minutes walk to centre.
The Alhambra B&B is nice, bit further out , probably 5-10 minutes walk.
The Norfolk lodge hotel (literally next door to me) 5 minutes walk from town centre
The mayfair is nice but is more suited to the older generation, has pool/gym.

The bus service is quite good, the number 15 goes from airport to liberty station (main bus station). Then there are a number of tourist site seeing buses which operate during tourist season, so durrel wildlife (aka Jersey Zoo), the war tunnels etc.

As to what to do in june/july (august youd be able to watch the battle of flowers which is quite good), I can recommend the following

War Tunnels
Durrell (must do!)
Maritime museum
Living Legend
The castles (Elizabeth, Gorey)
Opera House (go see a show)
The mansell collection if you like motorsport
Amazin Maze
Hamptonne Country Life museum

If you post what you like doing then I can refine the recommendations and provide costs etc.