Some utilities require more lead time than others to terminate service at your old residence and activate at the new. While the time required varies, you can use this checklist as a starting point.
One month or more before closing...
Cable Television and/or cable modem. Contact the cable company early, in case the new home needs any wiring or hook ups to accommodate your service. Call especially early if you are installing cable modem service.
Phone service. With today's often complicated home phone set-ups it is important to give the phone company some extra time - especially if you will need several extra numbers and/or jacks in specific locations.
BellSouth/ATT - 1-888-757-6500 (In state) 1-800-753-0710 (Out of state)
Satellite television and Internet service. Unless your new home has a compatible dish already in place, you will require installation services. Depending upon demand in your area this could entail a wait of a month or more, so put your order in early.
Fuel deliveries. If you receive automatic deliveries of heating oil or propane you should contact the supplier and terminate the service (and transfer the account to your new home if you still require deliveries).
Automatic deliveries. While they are not, strictly speaking, utilities, any automatic delivery services (bottled water, pool suppies, etc.) you may use should be contacted 1-2 weeks in advance of your move to terminate deliveries or change the service to your new address.
Several days to one week before closing...
Electrical service. Contact the electric company and switch your account to the new address effective on the closing date (make sure you have electrical service at both locations on moving day).
Gas service. Contact the gas company and switch your account to the new address effective on the closing date (make sure you have gas service in both locations on moving day).
Teco Peoples Gas - 1-877-832-6747
Water service. Notify the water company several days before the closing.
Seacoast Utility Authority - 561-627-2900
Sewer service. Notify the sewer authority several days before the closing.
Garbage pickup. If you pay for private garbage pickup you should notify the service provider a few days before the closing. Even if you have public garbage service at your old location you may need to arrange for private pickup at the new house. Check it out - you don't want to end up without garbage service just after you move in. If you want to have a special pickup - either at the old address or the new - make the arrangements at least a week in advance.
Satellite television and Internet service. 
Garbage pickup.