The subway may be a shithole, but it still is one of the greatest transit systems on the continent, and runs frequently enough where you don't need to check a schedule (unless it's like 2:00 AM, then I recommend checking a schedule because you might be waiting for a while).

If you're spending a day, I would try to plan it roughly around transit stops. Pick a few neighborhoods you want to see, maybe some restaurants / museums / whatever in them, hit a couple of them, then ride to the next neighborhood on your list. The grid nature of the city (at least, Manhattan) and numbered streets/avenues make it very easy to navigate and make judgements about the walking distance between places. The numbered streets run out in Downtown Manhattan though, along with the regular grid, and become names.

I don't go into the city very often, but "The Village" is a pretty cool neighborhood. It's the general area southeast of Washington Square and Union Square parks. This is where Webster Hall is located, and the former site off CBGB's, a notorious music club. There's a lot of restaurants and late night shenanigans in the area, especially around Saint Marks Place.