US 41...the Tamiami Trail...can be done by bike. Much of it has a paved shoulder, but there is a long stretch in the western section with no such provision. I've ridden parts...from Shark Valley east to Krome Ave.(Hwy 997). Traffic can be heavy as one approaches the Miami side of the Trail but the shoulder does help quite a bit.
Services are available but it's still a long crossing. It might help to check the wind conditions before attempting to cross.
Collier-Seminole State Park on the west side offers camping and primitive campsites should be available in the Big Cypress Preserve. For the latter, one could check with the rangers at the visitor's center on 41. Further east...food, water and restrooms can be had in the Shark Valley area. I'm just assuming that all of these services are back on-line after the hurricanes. The Shark Valley visitor's center is open.
Another option for services is Everglades City, approx. 5 miles south of the Trail on Hwy. 29. One would have to double back, though, to continue east.
As for reaching the Keys, it *might* be best to head south when reaching Krome Ave. It will lead to Homestead and should allow one to avoid much of the Miami sprawl.