You have three general options to reach Key West from the Fort Myers area:
1. Take a VERY short bike ride to Fort Myers Beach, then ride the ferry boat to Key West.
2. Ride south on Hwy. 41 (the Tamiami Trail) from Fort Myers. The Trail will eventually curve east across the Everglades region. At the intersection of Hwy. 41 and Krome Ave. (Hwy. 997), turn south and head for Homestead and Florida City. You can then pick up ACA's Atlantic Coast route to the Keys.
3. Follow ACA's Florida Connector route east to Lake Okeechobee, then south to Fort Lauderdale. Ride east to the coast and grab the Atlantic Coast route.
Peak tourist season runs generally from December thru March. You need to balance the traffic against the weather. The crowds thin out in the summer months - but the heat, humidity and tropical downpours can make for tough cycling.
Personally, option number 2 would present a more interesting trip than that offered by the Florida Connector route. Heading south from Okeechobee is a bit of a slog, followed by the traffic and congestion in cutting east across Fort Lauderdale.
The Tamiami Trail offers a number of campgrounds, the beauty of the Big Cypress Preserve, a taste of Everglades National Park, and the Homestead area.
Your mileage will, of course, vary.