Yesterday I took my girlfriend with me on a random 40 minute ride to stop on the side of the road to stargaze near a farm. It was a lot of fun being away from the suburbs. I've never been camping before but I'd like to get away just her and I as far as I can from civilization. Also information about equipment would be helpful.
In the southern part of the state, a section of the Florida Trail passes through Big Cypress. From what I remember, it's about a 20 mile stretch of trail where you won't see many others. Though it has been a while since I've been (12 years?), it was very remote (as FL goes) and I cannot imagine that it has changed much.
Camping in this area is seasonal, with much of it being flooded during parts of the year. During other parts of the year, it can be difficult to find water. There is a ranger station on the trail at the Oasis Visitor Center (on US41) at south end of the preserve which can provide detailed information on trail conditions and hazards:
Address: 52105 Tamiami Trail, East Ochopee, Florida 34141
If you've not done much camping, you should consider sticking to established campgrounds. Otherwise, you can easily get yourself into trouble and find that you do not have cell phone coverage. There are numerous campgrounds (some primitive, but still car accessible) in the area:
Since you also mention star-gazing, this area gets bonus points since the north end of Big Cypress is frequented by both the Palm Beach and Broward County astronomy groups as a "dark sky" site: