Newark is probably your best bet. La Guardia is a pain to get to and from both by Land and Air.
What do you have the most interest in or like to do? There is way too much in NY for you not to have a plan. If you like the Arts, Museum, Broadway, or something Historical. Different approaches should be taken.
If you want to See that statue of Liberty, book early on line. There are very few limited passes to get above the pedistal and into the statue. Security is beyond tight there. Ellis Island is included in the tickets.
Most Art Museums are free, they have a recommended donation and you can elect to not pay. Food is great everywhere but be prepared to pay $$$. I would avoid the bus tours but that’s me.
The Circle Line is a lot of fun. That's a boat trip that goes completely around Manhattan on a double deck boat, with someone on the PA system describing the sights. There is the Bronx Zoo, which is magnificent, and the NY Botanical Gardens right across the street. They have trams for the tired if you are too pooped to walk around the extensive acres. There is the wonderful Museum of Natural History, Moma art museum, various other art museums, Museum of the Moving Image, etc. Long Island has Jones Beach, Sagamore Hill, and a couple other terrific nature preserves. You can go whale watching in the Rockaways. Last summer, I saw 2 humpback whales and a great white shark! And you'll have to look up all the myriad of museums, of which there are scads. I can also recommend Ellis Island, where our grandparents came into the U.S. You get there on a fun ride on the Staten Island Ferry. Most of all, just enjoy yourself and be situationally aware.