I recommend Bivens Arm Nature Park for a picnic and some woodland hiking if you are in the area, but it is not something to go out of your way for.
To be sure, I don't want to come across as negative. This delightful park offers ample parking with quick access to grills, multiple covered picnic tables, emergency phone, and even wheelchair access restrooms. The trail is somewhat of a bonus, a place to wander during a family gathering while the others eat and talk. The park also serves a functional purpose in preventing or controlling erosion. Finally, there is a platform adjacent to the picnic area overlooking wetlands where others have spotted alligators, although in multiple trips I have not.
That said, the trails are somewhat aimless, mostly unnamed and unmarked. Some end suddenly in parking lots, or even people's backyards. You never get away from neighborhood sounds such as traffic, lawnmowers, and leaf blowers. Trails are narrow so expect insect swarms even by Florida standards. Wandering everything I could find, I got in about two and a half miles of hiking.