Definitely catch a broadway show if you have time (and the interest) - it's such a part of the NYC experience to me. If you visit one of the TKTS booths you'll get tickets to the show of your choice for that day/evening at a much reduced price. As one of the pp mentioned, there is a booth in Times Square, but it is often very crowded and you have to get there pretty early to get in line. Another, less crowded, option is the TKTS booth at the South Street Seaport. Here's a link for more info if you are interested
http://www.tdf.org/tkts/#.
I think what I love most is just wandering around the city. Pick the areas that you are most interested in visiting and just start walking. Great shopping, amazing restaurants, fun street fairs on the weekends (though I'm not sure about that at this time of year), people watching, etc. So much to do. I especially love wandering around in Soho, Tribeca, Little Italy, the East Village. So much to do and see (and eat and buy)! And if you decide to do Ellis Island and the Statue of Liberty, just go early as soon as they open. The lines are not bad if you are there first thing in the am.
Have a great time - you're going to love the NYC experience!