The best free thing to do down here doesn't cost anything and doesn't need a plan: walk the water. The southern tip of Manhattan is wrapped in a near-continuous waterfront path, and from 17 John Street you're only a few blocks from it in two directions. You can do the whole loop, harbor side, then East River side, in a morning, or break it into pieces over a couple of days.
We've laid it out as one continuous route heading south to the harbor first, then up the East River to the Brooklyn Bridge. It's flat the whole way, mostly paved esplanade, and there's shade and benches if you're taking it slow with kids or grandparents. Wear real shoes and bring water in summer, there's not much cover along the harbor stretch.