For food down town definitely check out Curly's diner on 14th between 2nd and 1st. Everything there is vegetarian and anything can be made vegan! Owners are awesome and so's the food! Other good places are, Red bamboo and vegetarian paradise near west 4th st. My favorite tea place is a Japanese one called Cha-an on 9th st. between 2nd and 3rd. They have awesome home-made mochi!!
Oh and check out the vegan potluck in nyc. It maybe happening while you are both here.
If you are looking for a more underground place this one is findable but at totally unique experience. You must go to Kebab Cafe in Astoria Queens. It sounds whatever but its like nothing else. The place is owned and run by one man every day named Ali. Occasionally he will have on helper but the place doesn't open without Ali there. There are about 5-6 tables in this place and no host not to mention no menus either. The dining room is decorated with lots of odd things from around the world. Ali will be your host your server and your cook, as well as your friend. He's an amazing guy with a great sense of humor. Basically, you tell him what you want. He will list things he is making, what he has that day. You can say vegetarian- specify no eggs no milk. The first time Ali cooked for me I cried. I am a chef as well and no ones food in a restaurant has ever struck me in such a way. Keep in mind you may have to wait a while for a table and outside too. Also, he cannot reproduce something he made before always. He makes stuff up all the time. I hope you go there and enjoy it as much as I always have!
Foodswings in Williamsburg (Brooklyn) is vegan comfort food. Always filled with hipsters. Definitely caters to an alternative scene.
I 2nd FoodSwings! I adore that place.
Dirt Candy in the East Village is quite interesting and experimental vegetarian food (can be made vegan of course). Pretty expensive, and really tiny, so make reservations!
Raw Food and Wine is also on the expensive side, but it's a necessary stop on your vegan food tour of New York. Restaurant is entirely raw, but somehow manages to be the most delicious food I've ever tasted.
Also, I give another vote for Foodswings in Williamsburg! Really delicious, and not even trying to be healthy. Vegan junk food mmmmm.
Viva Herbal Pizza on 2nd Ave and 12th street is good for vegan pizza. Not mind-blowing, but you can't miss pizza in New York, so this is a good stand-in.
-Angelica Kitchen on 12th Street and 3rd Ave.
-Vegetarian Dim Sum House in Chinatown.
-Stogo for Vegan ice cream
-'sNice for Sandwiches and salads. One on 8th Ave. and Horatio Street in Manhattan and another in Brooklyn.
-Taïm in the West Village (Waverly Place)
-Rainbow Falafel (tiny little bodega on 17th Street West of Union Square).
Other vegetarian places:
-Counter on 1st Ave.
-Red Bamboo and Vegetarian's Paradise 2 (Next door to each other on West 4th near Washington Square Park. Both pretty mediocre).
-Atlas Cafe on 2nd Ave. and 4th Street. The sandwiches and entrees are kind of hit or miss, but desserts are really good, from Vegan Treats.
-Quantum Leap (standard Vegetarian food. There are a couple different locations in Manhattan)
-Kate's Joint on Avenue B and East 4th Street is the same kind of vegan junk food at Foodswings, but not quite as delicious. Slightly more formal atmosphere than foodswings, but really it's just a glorified bar that happens to serve vegetarian food.
Good luck! Honestly, if you just wander around the East Village for an hour or so, you'll likely find yourself overwhelmed with the amount of vegan and vegetarian options. Feels like every other restaurant sometimes!