ooh you are staying in a great location, i used to live v close to there in Sussex Sq. ST James St is in Kemp Town and you can walk from there right into town. Kemp Town has some nice cafes, and places to eat. There's a lovely Thai place just on the corner of upper rock gardens (i think) its called Sawadee and its yum. Also a lovely pub called the Barley Mow, which i miss terribly and they do a fab SUnday Roast and have a little garden where you can sit out the back (so kids dont have to be in the pub).
As someone has already mentioned, Stanmer Park is lovely gorgeous place v green. There is a lovely little cafe if you walk right into the back of the park, they do lovely homemade cakes. You also have Preston Park which is nr London Road and again that is nice during the summer, huge open space v green.
ABout 20 or so miles out of Brighton there is Wakehurst Place, which is really nice. Botanical Gardens.
Try and have a look at The Brighton Dome's website as the always have summer work shops going on for kids, theatre, dance events etc.
Lewes is great, lovely place to visit, and if you do go make sure you go to Bills Cafe, as they make the most delicious food!!!!
And then there's the obvious; The Pier, marina, pavillion, lanes, oh and the lagoon towards hove, there are a couple of mini golf areas along the beach nr volks railway which always keep the kids entertained for an hour or so.
HAve a fab time!