February will be cold! Very cold and possibly snowy. April is better! However you should consider The Highline for a good walk, Brooklyn Bridge too. The Metropolitan Museum has something for everyone and is very strong on American art and culture. There is also the Tenement Museum in the Lower East Side. You can do Top of the Rock or The Empire State Building.
I am not sure how you get sports tickets. We were treated to a baseball match by friends but that was summer . It was also rather boring and 4 hours long! The audience ate, drank and talked all the way thorough.
You can get the Staten Island Ferry and also go to Ellis Island. Times Square is a wide bit of road with neon lights. As a first visit, stay central, but it's actually nicer in the SoHo area. More of a neighbourhood.
We went to a show with our friends and it was mega expensive so be careful what you book. There are outlets for unsold tickets like London. Just walking about is great and call into Grand Cental station for the ceiling and shoe shine boys. My DDs also liked Radio City Music Hall and so did I! It is a gem of Art Deco architecture.
If you do a bit of shopping research you may find a great store that caters for a particular interest or hobby. Americans are not snobby and you can walk into Tiffany and look at the diamonds if you wish. Happy planning.