Hi…we live in North Fort Myers. Harriet and her mate M15 (bald eagles) are back on their nest…google Fort Myers Eaglecam and it will show up. The nest is right on FL-78 Bayshore Road just east of the corner with Slater Road and about 2 miles east from the intersection of 78 and US-41 Tamiami Trail. Park in the Church of the Nazarene lot on the north side just east of the Publix shopping center.
Other good places…
Sanibel is really crowded and parking is scarce so I would personally skip it. A few miles north on 75 at the Tucker's Grade exit is Babcock Webb Preserve which has (if it's the right time of year) Red Headed Woodpeckers which are pretty rare. There's also the Venice Rookery up the coast a few miles in Venice, Myaka State Park up near Sarasota, and Corkscrew Swamp Sanctuary although all of those are a drive of most of an hour from Fort Myers. Further afield…halfway across 75 towards Miami just east of the Miccosukee Reservation Casino is Shark Valley. It is part of Everglades Nat Park and has a tram you can ride on or you an hike or bike the 14 mile paved tram path for birds and gators.
Myaka has some nice hiking paths, Corkscrew has a boardwalk so you stay dry…although in December the water might still be too high to allow maximum wading bird population. Venice Rookery is essentially in downtown and is a little island in a lake about an acre or two in size so you can get pretty close.
Then there's the "this is Florida, there is water everywhere and birds and gators are accessible just about any place. I was on my bike ride this morning…along US-41 in a sort of out of town but not in the boonies residential area…and in 8 miles saw a half dozen ibis, another dozen assorted herons mostly Great Blue and Great Egrets along the canal…also our local Swallow Tailed Kite, and an immature Bald Eagle along with a couple of hawks that were too far away to easily identify.
I don't know how much time you'll have to devote to birding…if not too much then I would do Harriet/M15 and Venice Rookery for highest probability of species abundance. Next would be Corkscrew (assuming it's open again, I'm not sure about their hours currently) and then Babcock Webb but it's sort of hit or miss and if the Red Headed are not there it's mostly hawks, buzzards, and songbirds.