The Bob Marley museum is a must see for any fan of the reggae icon, and is a great place to acquaint yourself with the man and his music. But Kingston also has some great areas to explore around the waterfront, and through the area is a bit dilapidated, it is still a pleasant place to wander from historic downtown. There are several markets that sell local art and crafts that you can browse through and these are a great place to bargain for authentic souvenirs of the island.
An absolute must see natural attraction is the Dunn's River Falls near Ocho Rios (the other main tourist centre) The falls are accessible after a steady trek through sweaty jungle, and once you get to them, the temptation to throw yourself into the clear ice cold water is easy to give into. Its one of the busiest places you can go to, but also one of the best days out to be had on the island.