Just click on an airport to go to its website:-

Southwest Florida International Airport in Fort Myers (RSW)  The closest airport to the house, just 20 minutes door to door, and in our experience very quick & efficient.  It has a reasonably new terminal, and is a very functional and pleasant place.  All the major US airlines fly into Fort Myers.  There is a direct LTU Airways flight from Dusseldorf, and there are constant rumours of direct flights from the UK.  There are good connections from US hub airports (Atlanta, Philadelphia, Chicago, Newark, etc).  If you are a European guest travelling to Florida in the winter and you are taking a flight that connects in the USA, you might want to consider the likelihood of your connecting airport being shut because of bad weather before choosing your flight / route.  Delta has the most southerly US hub, in Atlanta, so is least susceptible to snow.

Miami International (MIA)  If you want to use the closest airport with direct UK flights (only from London at the moment), this may be the one for you.  It's just 2 hours drive to Fort Myers, and it's an easy journey along the evocatively named Alligator Alley (the I75 interstate highway).

Another option for direct flights from the UK is Tampa International (TPA).  On the Gulf Coast, about 2 hours easy driving from Fort Myers.  Nice airport, good facilities.  Great drive down over the spectacular Sunshine Skyway Bridge across the mouth of Tampa Bay, with pelicans for company.

Orlando International (MCO) This is the main port of entry into Florida for many UK guests, especially those spending some time at the theme parks. Used by most scheduled US & UK carriers, including Virgin Atlantic and British Airways.  It's at the southern end of Orlando, closest to Kissimmee and to most of the parks.  Has direct scheduled flights from some UK airports.  Usually an easy drive down to Fort Myers, about 3 hours.

Sanford International  (SFB)  Used by most of the UK charter airlines that fly to Florida for a theme park-based holiday. At the northern end of the Orlando conurbation, 30-40 minutes from Orlando International.  About 3 hours away from the house.