We used a shuttle service to get from FLL to our hotel in downtown Miami. There are lots of them. We used Tri-Country Transport, they are based out of the FLL airport. It was $55 for a towncar to take us to our hotel in downtown Miami (there were 4 adults).
They have a website www.floridalimo.com
We rented a car from Thrifty at the FLL airport and dropped it off at the Hyatt Miami Downtown. Thrifty has an office there (plus we stayed there). I think the office at the Hyatt was only open until 6pm.
We have used Carnival out of Miami a few times, and flew into Ft Lauderdale.
You can buy a voucher from Carnival for pick-up at either the Ft Lauderdale or Miami airport.
We go in a day early and stay at a Ft Lauderdale hotel that has airport shuttle service, then just go to the Ft Lauderdale airport the next day and you will find the Carnival people at baggage pick-up. They will direct you where to wait for your respective ship's pick-up point.