Do you have an overnight stay in between or are you flying in day of (yikes)?
We've flown into FLL and sailed out of Miami twice and we did it two different ways. The first time we took a shared shuttle from FLL to downtown Miami, then took a taxi to the port in the morning. The second time, we rented a car, stayed in FLL and then drove to Miami, returned the car and took the car rental shuttle to the port.
The first way was logistically a little bit easier so I would take a shuttle. Here on the DIS we were recommended SAS Shuttle and they were awful. The driver was incredibly unsafe, the van smelled and was dirty and I will never use them again. I've heard good things about Jiffy Jeff so that's probably who I would use if I were to do it again. You could probably get an Uber, but you might want to price that out depending on how many people you have.