A decent network of trails exists, fanning out from around the Area Recreativa Doña Juana. The trail network is extensive, but complex, with paths in good nick but rarely marked. It’s best to plan ahead and ask about current conditions at the DRNA in San Juan, as mudslides are common.
It's best to consult with the ranger station for longer hikes. Out on the forest paths however, hikers may run into difficulty navigating with the basic maps available at the ranger station. They have photocopies of the reserve map marking key trails, however some of this material is extremely misleading. The trail map lacks useful detail and the compass rose has been rotated so north is not at the top of the page. You’re not likely to get anything better in San Juan from the DRNA, nor is it easy to get USGS maps on the island. Source a USGS map from map suppliers in the US, or mail-order one on the island.
Prepare properly: bring sufficient food and water, and insect repellent.