I wrote a very long post with suggestions, but then my browser refreshed, and I lost everything. Here's what I remember writing:
Midtown West/Hell's Kitchen- Hourglass Tavern (next to the Brooks Atkinson, lots of Broadway sightings, if that's your thing), B Side & B Squared (sister restaurants; brick oven individual pizza)
East Village - The Library (restaurant/bar in The Public Theater), Swift, The Wren, Veselka (famous Ukranian Diner w/ delicious, albeit overpriced pirogies). If you want brunch one day, go to Clinton Street Baking Company on the Lower East Side for the best pancakes you'll ever have.
HK & EV are both great areas, filled with lots of options. In both neighborhoods, you can get food in nearly every cuisine. Have a look at Yelp, or just walk around. Also, not sure how far east you'll be in the EV, but just a heads up that once you get into Alphabet City, it's a bit of a walk to the trains. The NR at 8th Street will take you straight to Times Square, but if you're all the way east on, say, Ave C or D, you'll need to leave yourself time (~20 min) to walk to the trains. It's very safe, if that's a question. Very different from the East Village depicted in Rent. I never felt weird walking around at night, but I also was a female in my early 20s during my time in the village, so it was a question for me.