The obvious things in DC are actually all pretty good if you've never seen them before. The Tidal Basin is a great looking place at night. The B&O canal is a pretty nice walk, especially where it overlooks Great Falls. There's a lot of stuff at the zoo and in the different museums that you really can't see anywhere else. Also, they're free. After growing up around DC, I'm always a little surprised when I go to other cities and have to pay for admission.
If you're a drinker, there's lots of neighborhoods to pub crawl with their own styles. Georgetown, Adams Morgan, around Dupont Circle, U Street, etc. Outside of that, if you love Asian food, I claim it's the best in America and possibly in the world. There's great Chinese in Rockville, Korean in Annandale and Vietnamese at 7 Corners.
As for New York. Don't see the sites. Just start walking. I always end up following the same plan. Pub crawl the East Village, free pizza at the Crocodile Lounge and keep wandering. 3 AM hits and it's yakitori or sliders in St. Marks, K-town for fried chicken, or congee in Chinatown. Falafel is for sissies.