"The Season" in Florida varies by which part of Florida you're in. I can only speak for the Treasure Coast, which is sort of the lower middle of the state on the Atlantic seaboard. Vero Beach is at what they term "the top of the tropics". That refers to what will grow here and our weather. In case you wondered, what can grow here is a jungle if you don't keep cutting it back!
People who come South for the winter are politely called "snow birds" (we won't go into what they are so impolitely called in some places!!!). Their stay varies by whether they own a home or are just renting. Most rentals are at least one month long, and many places require a 3-month lease. Vero Beach has few motels since the hurricanes of 2004 and 5, though there are several under construction. The community always fights any proposed motels on the barrier island ("the beach").
Home owners typically come down in October to reopen their homes and schedule any necessary maintenance. Then they return North for Thanksgiving and Christmas. Their arrival back in Florida can vary from December 26 through mid-January.
Three-month renters arrive as soon after January 1 as they can manage. Depending on where they have rented, they have golf, tennis and dining privileges until March 31.
A word to the wise: The absolute worst place to be in the world is I-95 (which runs the length of Florida on the Atlantic seaboard) the last few days of March and the first few days of April. If you ever rent in Florida, do yourself a favor and check into a motel until about April 4th before starting that long drive home. You'll thank me. I-95 becomes a parking lot during this time frame. Not a pretty sight. I'm sure the same is true of the Florida turnpike up the center of the state. Nuff said.
Other than March 31st, there are other trigger points for departure. Some people still return North to file their taxes. Others want to spend Easter up North. Others have discovered that the most delightful months in Florida are May and October. And if you check the temperatures in July and August, Vero Beach is often cooler than Washington, DC, or Richmond or New York City. Those ocean breezes really make a difference.