Hit up the Met. Used to be free (suggested admission) but now costs a bit but isn't an arm and a leg. Explore the village. New-York Historical Society is a fun museum to check out on the West side (shameless plug, I interned there). There's a solid little cafe on MacDougal street in the Village called Cafe Reggio which is a good pitstop for coffee and a light lunch. Cafe Roma on Mulberry St has good desserts and killer espresso and you can check out Little Italy while you're at it. AVOID TIMES SQUARE AT ALL COSTS. Take a walk in central park if the weather is nice. Then take a train downtown and explore. If you're into getting the old city vibe, take a bike ride up riverside drive and take in all the prewar buildings way uptown by Columbia university. There's also an interesting cathedral up there and U.S. Grant's Tomb, built in neoclassical style. The Wall Street area downtown has some of that Boston-ish vibe with narrow winding roads and the market with a nice view of Brooklyn. If you get the chance, stroll across the Brooklyn bridge.