There are many ways to see New York City. You can do it on a budget or you can spend a life savings. You can stay on the outskirts by the airports, La Guardia, JFK or Newark or you can stay in Manhattan.
Staying on the outskirts may not be as costly, but you're going to spend 1 1/2 to 2 hours each way in travel time. Do you really want to spend that much time on the subways & buses. I know I don't.
Why only 3 days? NYC is not cheap, though I will offer suggestions on Free Things to Do.
Most people are going to arrive in NYC by airplane. But if you are arriving by train, it will be in Grand Central Station or by bus to Penn Station. I would recommend you then take a taxi to your hotel.
Arriving in JFK or LaGuardia Airports, you have numerous options. In the past, I took the Gray Line Bus (approximately $17) to Grand Central Station and then I had to take a cab to my hotel. I have used Super Shuttle (www.supershuttle.com), located outside of baggage claim area, many times, but it's a shared van and it could take awhile, but it is cheaper and safer. They drop you off right in front of your hotel. To a Times Square Hotel, it would cost approximately $22. A hired can be booked for $75-150 one way. Uber or Lyft would be about $45 one way and a taxi is $60-75 one way.
Newark Airport is a little further with the same options, but a little more costly. I have done Uber for $45 one way (during non-peak time), Super Shuttle for $27 and hired cars for up to $150.
Now comes the hotel stays. I will list some inexpensive hostels, but these book up quickly and they are bare bone accommodations. Clean & serviceable, but nothing fancy.
Broadway Hotel & Hostel is near Central Park on the west side and has rates from $41 per night.
The YMCA starts at $109 per night
The Blue Moon Boutique Hotel is on the lower east side with rates from $49 per night
HI New York City Hostel is near Central Park with rates from $51 per night.
I'm not recommending AirBnB for NYC. Too many not great stories.
You don't want a hostel, OK I get it. Well, the best area for you to stay is in The Theatre District and Times Square. It's brightly lit and lots of police around constantly keeping the tourists safe.
Everyone wants to stay at the Hilton's, Westins and Marriot's, etc in the Times Square area, but you are looking at $425 a night and up, plus tax.
Once you get your hotel rate though, check it out how much the taxes become. If you look it up on the internet, it says a flat rate of $3.50 per day, plus 14.5% additional taxes. Don't believe that. It will be approximately 30% more per day for all taxes & fees.
Now typically other than hostels, the average moderate 4-5 star hotel will show $350 & up per night. For location and price, I would recommend The Paramount or Manhattan at Times Square. If you can wait until a few days before, if there isn't a convention in town, these same hotels will list for $125 a night.
Remember what I said about your life savings. That's what it takes to stay in NYC for a few days. Airport hotels are usually $158 a night and up, but then you have to take a bus to the subway and you have travel time.
Next you will need a Metro Card that can be purchased at the Hudson Shops at the airports. This is going to be your best way to get around for the next few days by bus or subway.
For those of you driving in, the parking cost is approximately $30-50 per day.
I would also recommend the Hop On Hop Off Bus pass. Take your 1st day of traveling around the city, seeing everything and then come back for more. This is a highlights way of seeing the city. You can also get off at every spot, if you want or just keep riding around and listen to the history of NYC.
Sights not to be missed: Statue of Liberty, Empire State Building (but there is a $50 entrance fee to go to the top) and the World Trade Center.
The most famous restaurant that I have been to is the Russian Tea Room, but I'm not going to recommend any others to you, because there is a restaurant every 3-5 feet (or at least it seems that way).
Do you want to see a few Broadway plays while you're in NYC? You can book your tickets ahead of time at www.broadwayworld.com or do what I do: On the day that you want to see a Broadway play; in the middle of Times Square is Time Square Tickets and buy discounted tickets on shows that are available that night. My average ticket price doing this is $75. Book in advance-$175 & higher.
Free Things To Do:
Friday evenings at the MOMA (Museum of Modern Art) is free entrance from 4-8 pm & every day from 930-10am, the Sculpture Garden is open to the public.
Take a ride to Brookly and visit their beautiful Botanic Gardens. It's free entrance on Tuesdays.
Ride the Staten Island Ferry for free every day!
Find the Whispering Gallery in Grand Central Station.
I have mentioned many companies, but no has paid me to promote them. I spend a lot of time in NYC, when I can afford it and I want to inspire you to go there too.