As stated above, Jamaica is a very dangerous place! DO NOT leave your group. Some poeple may be nice but most are money hungry. If you stay idle at one place long enough, some locals WILL offer drugs.
heCozumel.....well im not sure....i always take the boat ride to the main land for excursions. Cozumel is mostly but not limited to shopping. If you do go to the main land, keep a lookout. Its not anywhere near as bad a Jamiaca but kidnappings rarely DO happen. Again, Stay with your group.
The best place to visit is Grand Cayman! Like everyone else mentioned above, go on the stingray excursions awesome experience. This is the only place where i would say that its ok to venture out on your own. The locals are VERY VERY VERY friendly. The police don't even carry firearms
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If you want to go to seven mile beach just ask a local about the taxi. Basically theyre small vans with orange license plates in the front and backs that you wave down and ask to go to the beach. Its more like a city bus. They pass by every 3 minutes. They cost $2.50 per person, (pay when you're at the destination). Seven mile beach is the most beautiful beach i have ever seen!
Remember, the currency is like $1.25 American for every Cayman "Dollar". I paid $40.00 US for a fish and chips lunch for myself and the wife
HaveIf you need more specific info you can shoot me a private message.
Khem