Hi,
First, check out
http://www.happycow.net/
I've found if you ask for vegan food, and maybe suggest a vegan alternative to a dish on the menu (e.g. can I have the tomatoe pasta without cheese? Would it be possible to have extra mushrooms instead of egg? That kind of thing) most places will be happy to accomodate. Especially if you go to a locally run place, instead of a franchise burger bar or pub (though they will sometimes find you something, even if you just end up with salad sandwiches!). Smaller restaurants tend to make food from fresh, rather than having it all pre-prepared, so can alter it more easily.
Some cuisines lend themselves to being vegan more than others - I dread French-style restaurants (everything's dripping with butter!). But Thai and Japanese restaurants tend to have a few vegan dishes on the menu, and a lot of Chinese and Indian restaurants cater for vegans too. Italian restaurants and pizzerias can usually be persuaded to make tomato pasta dishes or cheeseless pizzas. You may have other world food (Ethiopian, Moroccan, Kosher) in NY, which can also be vegan. These types of cuisine tend to use vegetable oils instead of animal fats, though it's worth confirming (note - some indian restaurants will use ghee which is clarified butter, but others will use vegetable ghee).
Some places have started displaying a vegetarian or vegan society logo with their food - burger king, and some pasta bars. Or they will state if something has dairy content (apparently an increasing number of people are severely allergic to dairy?) The vegetarian or vegan societies in your country might publish guides to where to eat out.
The real pain in the *** is those working lunches or social occasions - but the joys of hummous and cucumber are slowly infiltrating even these! It's best to make sure catering staff know you're vegan early on (and, if they give you a frozen pizza with the ham picked off, quietly hint that you could sue

)
I guess how you feel about eating dairy is up to you. I originally turned vegan for health reasons - It didn't really hit me where milk and eggs came from until I started wanting children of my own (and then it grossed me right out!) So, yep, it's pretty shocking to me, but I can't say if that's the same for everyone
Sorry about the insanely long post - Hope that helps
