HI Candace and Ramona,
I live in the area and will try to answer your questions. The most cost effective way between the 2 cities is the train, depending on how much luggage you have and how adventurous you are. The train is about a few dollars for each person, plus a $10 cab ride from train to port of Miami.
If you need something easier, call 800-244-8252 for a shuttle service that is not too expensive and has great service. Not sure of the exact price for your route, but whenever I need a shuttle, I use them, and it's been the cheapest way to go.
Hotels: I suggest going on line and search hotels.com and cheaphotels.com. for the best prices. You can get a free shuttle bus provided by tri-rail at the FLL airport that will take you to the local tri rail station. You can stay right there at the tri rail station at the Courtyard Marriott. There are other Marriotts, so if you make a reservation, be sure and ask if they are the one at the tri rail station. In the morning take the train or the shuttle mentioned above (800-244-8252) to Miami.
Of course there's many other options for hotels if you want to stay at the beach, or in Miami, etc - depends on what your interests are. If you stay at the Marriott, you can also get a cab to the beach, or Las Olas (a shopping, restaurant and entertainment area) or a $1.50 bus to Hard Rock Hotel and Casino (which also has a shopping, restaurant and entertainment area)
If you take the train to Miami, this is the process:
free shuttle from airport to tri rail.
tri rail to Metro Rail Station in Miami
Metro Rail train to Government Center in downtown Miami
cab to the port (about $10 + tip for cab)
On return trip after cruise, when you are looking for something to do before flight. Depends if your cruise line will handle your luggage straight through to your airline, so you don't have to haul it around. There's plenty to do in Miami: for instance, cab to South Beach area for shopping, restaurants, entertainment, or just the beach, or go to Miami seaquarium, or the Vizcaya Muaeum (go online to check these things out, and other attractions, if you like).
If you have any questions about all this, you can PM me for my phone number and I'll be happy to help you out - Happy Cruising!!