The Staten Island Ferry
Free, runs 24 hours, and takes about 25 minutes each way between Lower Manhattan (Whitehall Terminal) and St. George Terminal on Staten Island. It passes close to both the Statue of Liberty and Ellis Island — arguably the best value view of the statue in the entire city, since the paid Statue Cruises boats don't actually land you any closer for most ticket tiers.
Do you need to get off the boat?
Not necessarily. Riding over and immediately riding back is completely normal and gets you the skyline-departure and statue views both ways. If you do want to explore, St. George has a small but growing waterfront area with a minor-league baseball stadium and views back toward Manhattan.
Beyond the ferry
Staten Island is the most suburban of the five boroughs — more single-family homes, more green space per resident, and a noticeably slower pace. The Staten Island Greenbelt offers hiking trails that feel a world away from the rest of the city, and Freshkills Park — a former landfill being transformed into what will eventually be one of the largest parks in New York City — is a genuinely unusual long-term urban project worth knowing about even if it's still mid-transformation. The St. George neighbourhood off the ferry terminal has the restored St. George Theatre, Snug Harbor Cultural Center, and a minor-league ballpark with a Manhattan skyline view worth stepping off the boat for.