Each day below gives you the area, the order of the stops, how you move between them, and what to do when the weather turns. Read the times as a frame rather than a schedule you have to hit.
Day 1: Lower Manhattan, the Statue of Liberty and Brooklyn
Start downtown with the Charging Bull near the bottom of Manhattan. The visit is a photo and a lap around the bronze, so half an hour covers it if you arrive before the queues build. From there, walk toward the water for the harbor cruise out to the Statue of Liberty and Ellis Island. Give that half a day: the boat ride, the security line, and the walk around the island eat the morning, and the indicative ticket for the Statue of Liberty cruise is $48.
Back on land, One World Observatory puts you above the same skyline you just crossed, with a ticket the same page lists at $54. Next door, the 911 Memorial and Museum is the part of the day that takes longest if you let it, and it is indoors, which makes it the rainy-day anchor for this stretch of Manhattan. Chinatown and Canal Street are a short walk north for a late lunch.

Finish with the walk across the Brooklyn Bridge, one way, then look back at the skyline from Brooklyn Heights or DUMBO before taking the subway home. Walking the bridge is a classic for a reason, and going at dusk means you catch the towers lit on the return view.
Moving around: the subway and the NYC Ferry cover this day. If your legs are done after the bridge, ride back rather than walk it twice.
Day 2: Midtown from East to West
Begin at Grand Central Station and watch the morning rush move through a working station. The United Nations Headquarters is a short walk east, and the flags and the river view from outside the complex take about ten minutes.
From there, work west. The Empire State Building observation deck is listed at $29, and the climb north to Rockefeller Center adds another deck at $31 if you want a second angle on the same skyline. Pick one deck for daylight and save the other for another evening, because two observation floors in one afternoon blur together.

The Intrepid Museum sits well to the west on the waterfront, an aircraft carrier the page lists at $38, so plan it either first thing or last rather than wedging it into the middle of the walk.
Midtown closes at Times Square and the Broadway Theatre District. The new Broadway season opens in autumn, and a show is the one booking worth locking down weeks ahead. If you are here in winter, the Rockefeller Center Christmas Tree and the ice rink at Bryant Park's Winter Village turn this evening into a different kind of stop.
Moving around: this day is walkable end to end, with one subway hop if the Intrepid is on your list.
Day 3: Central Park, Museum Mile and the Villages
Spend the morning in Central Park, which turns red, orange, and gold in October and stays the city's most forgiving place to walk. Come back along the Upper East Side for Museum Mile, then pick one museum rather than two. The Met sits on Museum Mile, and MoMA is a shorter ride south if modern art is more your line, and either one will hold you for two to three hours.
After lunch, take the subway downtown to the High Line, a short elevated walk that drops you down toward the riverside, and finish the day in Greenwich Village and the West Village. That pair is where cobblestone streets and jazz clubs sit side by side, which makes it the right place for a last dinner without a schedule behind it.

Moving around: one subway ride covers the jump from uptown to downtown, and the neighborhood itself is for walking. The late finish costs you nothing here, since the subway runs around the clock.
Rain swap: trade the park for a museum morning, move the High Line to a clear evening, and keep the villages for dinner, whatever the sky does.