I am visiting NYC in April with my husband and 14-year-old for 5 nights. I am looking for a neighborhood to stay other than Times Square. Somewhere we can walk to eat or shop. Neighborhoods I was looking at were Soho, East Village, Greenwich Village, Upper East Side. Thoughts? Budget is around $600 per night.

Here are some of the things we plan on doing which on our trip.
Broadway Show
Yankee Game
911 Memorial
Statue of Liberty
Ellis Island
St. Patrick's Cathedral
Rockefeller Center

Personally I like the Chelsea area as a base. Easy walk to the Hudson River, High LIne, etc. Lots of chain hotels that will fit your price point, restaurants etc. North/south subway lines on nearby Avenues, so easy to get around. From there, the Bronx is up and the Battery's down.

You can also look at places in the Lincoln Center area or areas on the West Side, north of Times Square and north of Midtown.There are also lots of chain hotels in and around Herald Square so it depends what you are looking for but you should book soon as if it is school vacation week or Easter/Passover break, NY will be a busy place.

The hotel we stayed in is closed, but the Columbus Circle area metro was convenient. Others can say about current hotels.

We stayed at the Warwick Hotel a few weeks ago and enjoyed our stay. It is walking distance to several of the places you mentioned: Rockefeller Center, St Patrick's Cathedral and Broadway shows. It is also close to Central Park where we took several walks. This is an older property but clean and comfortable and relatively reasonable (NYC is not cheap!) Buses and subway stops were convenient. We thought the hotel's breakfast prices were high so walked to other nearby locations for breakfast.

I recommend trying to get tickets for the first boat of the day when you visit the Statue of Liberty and Ellis Island. The security lines were long when we visited a few years ago but not for the early visitors.

Chelsea is great - convenient to transit, walkable and with a fair number of independent restaurants and shops. I also like staying way downtown in the area near the Financial District and the World Trade Center. If you stay down there, there are lots of great paths for walking - to the west is the Hudson River, just south Battery Park and easy access to the ferries for the Statue of Liberty, Ellis Island, Staten Island, etc and the South Street Seaport and Brooklyn Bridge are within range for dining and walking. The Millenium Hilton should be within your price range and I believe they recently remodeled their indoor pool area, a rare amenity in Manhattan.

We stayed at this hotel, in which we had a lower priced room with no view and little space around the bed. But the bathroom had been renovated. It's location is ideal, just one blck from the West 4th Street subway station, which includes the ABCDEFM lines.

Look in the East Village for a less touristy area. Chelsea, Midtown, etc you will be with other tourists. Can’t beat walking and restaurants in East Village. Now that Airbnb is not legal in NYC there is not much lodging in E Village so look immediately south of Houston in the Lower East Side.