First, I got the heck out of the midwest and moved to a state with sufficient public land, first to Arkansas and now here in rural New Mexico. Not only do we have abundant public land, but the human population density is low enough that strict visitor control isn't necessary. No-trace backpacking and backcountry camping is allowed on the vast majority of land, off-trail hiking is the norm, and permits aren't necessary except in rare cases (National Parks for example).
Second, the BLM land management maps are an incredible resource. They show national forests (USDA), national parks and monuments (US Dept of Interior), federal wildlife refuges (US Fish and Wildlife), state parks, state forests, state wildlife areas, national grasslands (USDA), Army Corps properties (US Dept of Defense), and BLM land (US Dept of Interior). They are nowhere near as detailed as 7.5 minute topo maps, but for general navigation showing property lines, major drainages, and roads they work pretty well.
If visiting a particular national forest, stop at the forest service ranger offices for a forest-wide map showing roads, general property lines, and major geographic features. There are also sometimes more specific maps for each ranger district that show hiking trails, scenic spots, and better road details (knowing ahead of time which roads are vehicle accessible can save hours of frustration).
Third, when on public land (or a guest on someone else's private property) I follow a strict no-trace ethic. Fires are burned to white ash, fire pits are filled in with the sod replaced, and living trees are never marred - for examples. Artifacts and fossils can be photographed but never moved, looted, or destroyed. And I never camp within sight of a trail or road - for my own privacy and security, but also as a courtesy to other visitors.
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I'm unfamiliar with New Jersey as far as outdoor recreation. But I know there are state parks, some of them large enough to provide some opportunity for backcountry exploration. Check Google Maps for other types of parks and public land: county parks, wildlife areas, city properties.
And of course if you're in New Jersey it's a given that you aren't far from several other states, states that may provide better opportunities. Pennsylvania, New York, West Virginia, Vermont, New Hampshire...
Looking at the New Jersey map, something I've maybe never done before, I see a lot of public land and parks. And some of them look large enough to keep a person busy for weeks. Wharton State Forest, Delaware Water Gap National Recreation Area, Great Swamp National Wildlife Refuge...