Hi lisa-1965, Welcome to Cruise-Chat!
Sorry I can't help with Montego Bay, as I've never been there. Grand Cayman is easy, just take a taxi up to the Seven Mile Beach area. It's not very far and some of the hotels have day pass type deals that include beach chairs etc.
Also check out our Caribbean Port Section located
HERE. There is a wealth of information there.
Welcome Aboard,
Dwayne