Carnival has several 3, 4, and 5 day itineraries out of Florida. You'll pay less than $100 per person per day for one of those cruises, sometimes much less. IMHO Carnival also gives the best value.

There are no major cruises to Bermuda from Florida. 4 and 5 day itineraries will generally take you to either one or two ports in the Bahamas, Key West, the Caymans, Jamaica, or Cozumel.

My personal recommendation for a 5 day cruise from Florida would be Carnival Inspiration out of Tampa. She's an older ship, but run very very well by an excellent crew. For the family she's got the Waterworks water park on the aft end of the top deck. The only balconies are in the suites, but they're not ridiculously expensive. Every two weeks she does a Saturday 5-day, a Thursday 4-day, and a Monday 5-day. The 5-day cruises both go to Grand Cayman and Cozumel. The 4-day goes to Cozumel, but IMHO there isn't any savings doing the 4-day vs 5-day, so the only consideration is the extra day off work.

Tampa is also generally regarded as the smoothest embarkation and debarkation port in Florida.

We've sailed Inspiration 3 times and loved every trip.

The only cruise line I would actively avoid is Celebration Cruise Line.