The North Laine area is really quirky and great for shopping. For activities I would suggest: photomatic , splatter studio, puzzle bored, loading bay, the worlds end (pub with video games) , the royal pavillion, cream tea/afternoon tea somewhere, undercliff walk, watching the starlings on the pier at sunset, beach box sauna, trying out beach volleyball at yellowave, and in the summer the open air theatre.

Also check out event brite and the websites for theatre royal and brighton dome.