The hand shake is the hook.. if you start with that you'll get the 'cool' handshake and likely more, a pitch, a 'gift.' Another greeting is for a higgler to offer your 'pretty lady' a gift where the solicitor ties a wrist band on her, don't let them! You may be on a walk and someone will join you and just start chatting it up where 20 mins later or so they may want a fee for the tour. The list goes on.. The regular greeting on Jamaica is "Wa gwan" akin to "what's going on".. this young lady is really cute and put a great piece together on greetings: https://youtu.be/ic5O-8GiOg8
A local man should have permission from you to speak with 'mi lady'.. well, it's really more lax than that, but you can call someone on this.. at least I have. I had a guy hassle us on the beach one night and I told him he was disrespecting me by speaking to 'mi lady' without my permission.. that was pretty much the end of discussion. I would say do not leave your lady alone at a local bar without a friend being with her.. as soon as you step out she will be approached and will hear some pretty crude and frankly down and dirty lines.. Please do not characterize all Jamaicans by my points, most Jamaicans are absolutely wonderful, friendly and warm people especially given the economic and political oppression they deal with.
If someone asks you for a drink I would suggest you tell them that if they buy the first round you will buy the next two rounds and see what happens. Here's hoping you have a great time.
-Seamuis