When I visited the Big Apple I found that living there is quite expensive and so is visiting there.
If you have a friend or relative you can stay with, this can save you quite a bit on accomodation. May I suggest that you avoid staying with them too long and that you offer to treat them to lunch or dinner, buy some groceries and help clean up their home.
If you're not favored with having friends or relatives who can put you up, there are other options which can still help you save.
You could try couch-surfing. I've heard a few people have had some success with this. I didn't feel totally comfortable using this but I may risk it in the future.
You could also try a service like Air B n B. This is what I used and it worked out well for me. There has been some conflict about this service recently in the news and legally, so double-check before you book accommodation there.
Warning: I wasn't charged a service charge the first time. But my experience was that they will start charging you a service charge when you return to use it again. Make sure you are very clear about any extra fees you might have to pay to your host(s) such as those for cleaning or taxes.
Hotels are far too expensive for someone traveling on their own. Finding a more economical option means that you have more money to do other things like sightseeing, shopping or dining. Having a place that's clean & safe is very, very important.
If you are planning to do some shopping, know that clothing under $110 in value has no sales tax added at the register.
This is quite a newer rule in New York. In the past, many NYC residents would go to NJ to avoid having to pay sales tax on their purchases. I think that this makes it easier for small businesses to sell locally- made items too such as the ones I found in Chelsea Market.
If you go to Chelsea Market, you'll see that it's also an access point for the Highline Park which used to be an elevated railway. Now it's an elevated green space with trees, plants, pathways, spaces to sit and the occasional leftover bit of old railway. It reminds me a lot of the Couloir Vert in Paris which also used to be an elevated railway.
There are also lots of activities that you can do for free or very little money.
For $29, you can get a Metrocard with 7 days of unlimited travel on the Metro in NYC. This includes the subway and buses. This is one of the best public transit deals I've found in a North American city. Here's a link to the page which gives you more information about the different Metrocards offered.
The Staten Island Ferry is always free. It offers you views of the harbor and you can also see the Statue of Liberty. If you want to actually get closer to it and climb it than you'll have to pay.
Of course if you really like to walk and don't want to go much farther than the area where you're staying, you could walk. When you walk you get to see so many things you might otherwise miss.
There are lots of green spaces in NYC besides Central Park.
Just walking around NYC is very interesting and there are lists of interesting things to see.