If you don't have a car, walk or bike across the bridge and turn south immediately upon making "landfall" in New Jersey. First you come to historic "Fort Lee," which has recreations of Revolutionary War dwellings of soldiers. The next entrance puts you on Henry Hudson Drive, which winds its way north along the river below the cliffs. This road is shared by bikers and cars, so be careful. Hikers will want to get off this road and take the trail that runs north along the river itself. From this trail you can get right up to the river, observe wildlife, and enjoy some peaceful sandy beaches. The trail is rough in spots, so bring good shoes and good sense of balance. Look for sea glass in the river and historic bottles along the trail (I assume tossing bottles off the cliffs was a favorite pastime in the 1700s and 1800s). If you venture up to the walking trails that run along the cliffs, be careful with young kids along the edges.
If you have a car, you have a few additional options from Hudson Heights. Take the Palisades Interstate Parkway north and pick one of the exits that suits you. Visit www.njpalisades.org to obtain maps. There are 3 maps, each covering about 4 miles of the park. I highly recommend this park in the fall, particularly October and November.