How far do you want to ride on your first day? Stealth camping in urban Ft. Laud.-Mia.
area is all but impossible, and dangerous. I would not recommend it.
It's about 60 miles from Ft. Laud. to Fl. City. If you can do that in a day, you should have no problem finding a place to stealth-camp if Fl. City. If you want an easy first day, Oleta River St. Park is about 15 miles south of Ft. Laud. They have cabins for $55.
Personally, I would probably stealth-camp in this park. Big park with lots of wooded areas. About 15 miles south of Fl. City is a private campground(Pelican Cay) with tent sites for $35 for 2 people. This is on the stretch of US1 between Fl. City and Key Largo.
If you don't plan on camping there, take Card Sound Road from Fl. City to Key Largo instead of US1. It's a little out of the way, but there will be much less traffic.
Going into the Keys, you should have no problem finding places to stealth-camp, especially north of Islamorada and south of 7mile bridge.
A hammock would be ideal, but you'll be just fine in a tent, it's the dry season right now. A dark colored tent(olive,brown) would be your best bet for stealth.
Heres something I used to do back in my 20's that you might want to try...or not. I used to take a lot of road trips down to the Keys with my girlfriend, when I was young, stupid, and poor(God I miss those days!). Wanting to spend our money on alchohol rather than room-service, we would sleep on a concrete slab(air mattress is mandatory) that a house was going to be built on. In the keys, houses are built on stilts because of flooding, so the slab was 10 feet off the ground. This made for a nice breeze, which helped big time with the mosquitos, and the view was awesome. We did this dozens of times and never got caught. Just leave at first light and you'll be ok.
In Florida, you cannot be arrested for tresspassing if it is not posted and there is no fence. DO NOT go into a vacant or abandoned or hurricane damaged structure, that is a whole different scenerio.
We would sneak into a resort or rv park to take showers. A lot of hotels have showers in the bathrooms by the pool area.
For the most part, people in the Keys are very laid-back and friendly. If you talk to enough people, you could very well get invited to share a campsite, pitch in someones yard, or even stay on a boat for the night.

Things to see...
Besides the obvious Tiki Bar-Hemmingway House-Mallory Square at sunset(all worth seeing), two places come to mind that are a must-see. The first is Coral Castle. It is about 5 miles north of Fl. City, before you even get into the Keys. It was built single-handedly by a man who was broken-hearted over a girl. He said he discovered the secret of how the Egyptians moved such big and heavy stones without machinery. The place is truly a wonder and worth checking out.
The second is a place called Robbies. It is just south of Upper Matecumbe Key, which is just south of Islamorada. Over the bridge on the right. Basically, it is a restaraunt/tourist trap, but the reason people go there is to feed the tarpon. In the water, hanging out at the docks, on any given day, is over 100 tarpon, some weighing up to 200lbs. It's quite a sight. The coolest part is that you can buy a bucket of baitfish, take a bait and hold it 12 inches over the water, and a tarpon will take it right out of your hand.
I live in Pompano Beach which is about 5 miles north of Ft. Laud. You are welcome to camp in my back yard on your first night if you wish. It is very tropical, and I have a big fire pit. We can bar-b-q and talk touring. I don't know much about couchsurfing.com, but as a fellow tourer, I have over 5,000 touring miles under my belt. I think that qualifies me. P.m. me if your interested.

Word of caution. Be prepared for some of the worst, rudest drivers you will ever encounter. Hit and runs are a common occurence down here.

Good luck.

BTW- There are no alligators in the Keys. Sharks yes, alligators no.