This spreads the city out more sensibly than a single frantic day, grouping neighborhoods so you're not crisscrossing Manhattan.
Day 1 — Downtown & the Harbor
Morning at the 9/11 Memorial (free) and Museum, then walk to Battery Park for the free Staten Island Ferry round-trip past the Statue of Liberty and Ellis Island. Afternoon in the Financial District (Wall Street, Trinity Church) before walking across the Brooklyn Bridge into DUMBO for sunset skyline views and dinner in Brooklyn.
Day 2 — Midtown & Museums
Morning at the Met or MoMA — pick one, both are large enough to fill a morning. Early afternoon: Top of the Rock or the Empire State Building for skyline views (book a timed slot in advance). Walk through Rockefeller Center and Fifth Avenue, then Times Square in early evening before a Broadway show — buy same-day discount tickets at the TKTS booth if you're flexible on show choice.
Day 3 — Central Park & Upper Manhattan
Morning walking or biking through Central Park (Bethesda Terrace, the Mall, Bow Bridge). Early afternoon at the American Museum of Natural History or a Museum Mile stop on the Upper East Side. Evening: SoHo or Greenwich Village for dinner and a wander through Washington Square Park.