The best way is to take a ship-organized tour that gets you out of the city.
I do NOT recommend that you walk into the city on your own. Jamaica is notoriously dangerous to outsiders, especially if you look like a tourist.
But you could take a tour to a Jamaican banana plantation, for example, and see the old colonial style of life there.
Or if you go to a small town you will most likely be safe - but HAVE A GUIDE with you (hire one) and be cautious.