I don't think much can go wrong for you in Island Time. Perhaps I'm too casual! The crime rate is very low so I seldom lock my car or house and I'm generally comfortable letting the kids play mostly unsupervised while I enjoy a book. Be careful with the sun, particularly in summer. A bad dose of sunburn will spoil your time with us. Wear a hat and use plenty of lotion and sit under a tree. Drink plenty of water and don't get heat stroke.
Marine stingers can be prevalent from November to April, particularly from January to March - we mostly still swim when the tide is going out and there isn't a wind rolling waves (and therefore stingers) onto the beach. There is an element of risk in this, though, so be your own judge. Stinger suits are readily available for hire and protect against stingers and the sun - not a bad combination.
Sand flies really get me - I usually don't notice them until the next day when I'm covered in bites. When you're on the beach, use some repellent, particularly in the morning, late afternoon and if it is raining. March flies can be bad (and that's an understatement) generally only in November/December, but also on the odd other occasion. I have found Bushman repellent to be best for these - if you don't have this, sit in the water.