Check the dairy-free and vegetarian sections of the supermarket, but check the labels very carefully
. Some vegetarian food will contain milk (in some of its various forms - not just milk but whey, lactose etc) or egg (as this makes a very good binding agent). Equally not all dairy-free food will be appropriate either, again it is a matter of reading labels. Quite often you will see little logos stating suitable for vegans, which are always good to look out for.
Kosher Delis are also good places to find meat and dairy free products, but again be careful as what is permitted under Kosher definitions of dairy free or meat free will not correspond with the vegan definition. (for example, something may contain fish gelatine, which is Kosher, but not Vegan)
I am also aware that many US vegans will not eat certain types of processed white sugar, although it is OK in Britain, where I am.
Dairy free margarine is very good for cooking, vegan friendly jello crystals which is set with carageenan (seaweed derivative) instead of gelatine (made from animal hooves!) always go down well with children as do Swedish Glace and Freedom non-dairy frozen dessert instead of ice-cream. Seriously, I like my icecream and I was very impressed with that stuff. ETA- sorbet is often suitable as well, but check for egg products. Linda McCartney foods are often suitable as they don't tend to have egg or milk products in, unlike Quorn which often does. Their sausage rolls were particularly nice last time I had them.
Vegan friendly food can be anything from vegetable chilli, to mushroom risotto, eggplant casserole, vegetable curry or a simple pasta in tomato sauce, to rhubarb crumble served with vegan ice cream (see above)
In a pinch, just cook some pasta, and make a simple tomato sauce to go with it. Just leave off the parmesan!
If you are catering a party for either adults or children, and have a vegan guest, let me know and I'll dig out my menu.
This is a link to a UK website with some helpful info, although it may not have all the products available in the US, it can provide a good startup guide about what to look for http://www.vegansociety.com/