Silver Springs State Park offers glass bottom boat tours, kayaking and more at the main entrance on East Silver Springs Boulevard, but your best entrance for hiking is on 1425 NE 58th Ave, FL 35, mapped below.
It's amazing how close to town you can be while still feeling away from it all for hours. Trails include Sand Hill Trail near the entrance, Sink Hole Trail with plenty of signs to help you learn about and recognize local flora and fauna, and the River Trail and Swamp Trail which meet great views at Silver River. The views range from hardwood hammock one moment one moment opening into longleaf pine the next, distinct marsh and swamplands, with birding at the river always a short distance away.
This is an easy day trip in town, with plenty of visitors to interrupt the solace, but don't let that turn you off from how well preserved and beautiful the land and sites are. Silver Springs State Park is truly a gem of Ocala.