Reading the label is a very good idea. Sax mentioned the allergy info which is a good double check. But I'm not sure that info is totally relevant. I think the key word in that "warning" is "may". I think "may" is just a way the company covers their a$$. Like if they have a plant that processes nuts or dairy or something in the same room, there is a chance that the product could be contaminated with something. but I wouldn't call that non-vegan. just a risk if you are highly allergic to something.
There are also some keywords you have to look for. I still can't remember which red dye is made from beetles. And I think gelatin and lecithin are sometimes vegan and sometimes not. So that doesn't always work for me, either.
Dairy can be listed as any number of things but most of them are obvious. Although when I first became vegan I didn't know what whey was.
there are some pretty good apps that can help you if you are in doubt. The one on my phone is called "IsItVegan?" I know there are a few others. With IsItVegan you just use the phone's camera to scan the barcode. But it's not foolproof either.
There are also apps that can tell you if a product is cruelty-free. "Bunny-Free" is the one on my phone. And I know that there are others. I think PETA, CrueltyFreeKitty, and LeapingBunny all have searchable databases.