At the very least treat it heavily. The Trail Conference has a brochure covering exactly this question. Many smaller children like steep, rocky trails or ones along streams because there are more interesting things to explore in detail. Take your time and move at their pace. The Trail Conference sells two books specifically about hiking with children.
Many parks allow dogs on a leash, but call ahead to be sure.
Many of the phone numbers you will need are in the New York and New Jersey Walk Books. Parking areas are shown on the Trail Conference hiking maps. The Walk Books have explicit instructions on where parking is available. Unfortunately the parking availability changes faster than one might hope so have some alternate plans - a good idea anyway because the parking lot may be full when you get there. The Trail Conference sells hiking maps of many local areas.