Wow, that course is insane! But what fun! I think you'll get an idea of what neighborhoods are interesting to you, from running the course.
Learn the subway system-- it's easy and cheap and fast. I don't know what you're into, but I tend to like museums and hang-outs. Major museums run the length of Central Park's east side; the natural history museum is across the park on the west side.) The Lower East Side has several vegan businesses and a tenement museum. Washington Square and Union Square are fun hang-outs.
It's touristy, but the Grey Line double-decker bus is a great way to see the city for the first time. You get all of your "must see's" (Empire State building, Flatiron building, etc.) taken care of, and you can get off and walk around and then get back on later.