You could spend months exploring every option. Mount Tam has over 50 miles of trail within the park and connects to a larger, 200-mile-long trail system. If you felt like it, you could go from the summit (2500+ ft) down to the ocean. Many people run this on the Dipsea race!
However, for a less extreme adventure (about 4.5 miles of hiking with moderate elevation changes), try parking at the Pantoll ranger's station and doing a loop hike up to the 3 peaks and then around the back. You'll get to experience amazing vistas and quiet groves and peaceful alpine meadows.
Start out on Old Stage Road, which was a railroad that led up to the historic West Point Inn. This little mountain retreat is still in operation, and is only $30 to stay the night. The accomodations are pretty rustic-you need to hike in with your sleeping bag, but that's part of the charm.
You'll next take Rock Springs trail up. And I mean straight up. This is the hardest part of the hike and on a narrow little trail that could use some maintenance to boot- but worth it for the views. You'll end up on the access road to the East Peak, about .1 miles from a very conveniently located water fountain and restroom.
Take the little extender trail from the parking lot to the summit. (As you can figure out, non-hikers can drive here.) The views from the top cannot be described in words other than "wow."
You have several options at this point, but I'd recommend heading out to Old Mine Trail. You'll pass a lovely alpine meadow and can take a detour to the Mountain Theater. Not much else to report, except more lovely views of the Pacific as you head back towards where you parked at PanToll.
There's also water and restrooms at PanToll, and be sure to pick up a map so you can choose your own routes and be able to navigate.