So far, this is what we have as our plan, but your recommendations on the East & South East Florida appear quite attractive.
2 days: {Merrit, Vierra Wetlands, Lake Apopka}
4 days: {Fort De Soto, Ding Darling, Corkscrew swamp sanctuary area}
2 days Big Cypress/Shark Valley & Flamingo area
Last day before boarding flight, possibly through Wakodahatchee Wetlands?
Since you mention you're really just starting out with birding and don't have a 'hitlist', I don't think you need to worry as much about trying to fit in east and west coasts - the birds you'll find on both sides mostly overlap but for a few oddball things that might be more prevalent on one side or the other. There are two types of birding spots common to Florida:
1. The first type is the small, densely packed birding hotspots that are usually rookeries, with birds that are quite accustomed to seeing people and allow you to get very very close. I think you may be better off focusing on these types of spots for your first trip - hit just a few of these hotspots and you can fill your 'lifer' list to nearly 100 species in just a few hours at each spot. It's a good way to get started with birding and get a ton of species. Wakodahatchee Wetlands, Green Cay Wetlands, Ding Darling, Viera Wetlands, and Anhinga Trail would be examples of these types of spots - where you can get very close to birds and where you don't need to do a lot of searching or walking around to find dozens of species. At Wakodahatchee this past Saturday, in 1/2 mile from 4:30pm to 5:15pm, and shot 37 species, 20 of them in flight and sitting.
2. The other type of spot is a more wide open area, with birds spread out, often a little more skittish, requiring much more walking and close observations to get the rewards. Often the birds in these spots can be very dependent on time of day, with dozens of species in the morning and none in the afternoon, for example. These places can reward with some cool species, but you often must cover a lot more ground, and spend a lot more time, to notch 6-12 species, and often you'll need good strong binocs or a big telephoto lens, as the birds don't always let you get as close. Arthur Marshall, Fort DeSoto, Lake Apopka, Corkscrew, and Big Cypress would be examples of this type of spot.
Personally, I think you'd do just fine to stick from central east Florida down the east coast through S. Florida and down to Everglades/Miami area. You of course have other things you can do aside from birding if you choose, from theme parks to beaches...but that would let you hit some really dense populated birding spots like Gatorland, Merritt, Viera, Wakodahatchee, Green Cay, Key Biscayne, and Anhinga Trail/Everglades...and with any free time in between, there are dozens of smaller hotspots and wetlands parks you could add in along the way (Orlando Wetlands, Winding Waters, Grassy Waters, Peaceful Waters, Wellington Wetlands, Evergreen Cemetery, the STAs (Stormwater Treatment Areas), Markham Park, and so on.
If you decide to go west coast, you could put together a similar itinerary - so I'm not endorsing one over the other...I just happen to live on the east coast so I'm more familiar with what's on this side of the state.