If you've not been to the Royal Pavilion in Brighton, Caran, I strongly urge you to pay a visit:
http://brightonmuseums.org.uk/royalpavilion/
While everyone seems to know about The Lanes in Brighton (which are full of over-priced designer shops but some quite nice, reasonably-priced bistros too), many people still miss the wonderful North Laine area (which, in my opinion, offers far more to visitors seeking out unusual shops:
http://northlaine.co.uk/
Central Hastings is a bit of a dump but the Old Town still has quite a lot to offer. (Walking along the seafront, with the sea on your right, keep going until the road heads inland a bit. Follow it to find interesting shops, bars and bistros).