Experience and time is all it takes. Mistakes will be made, but they shall be overcome.
There's a list of animal ingredients on my website that people refer to really often. You can start there. http://www.vegfamily.com/lists/animal-ingredients.htm
Also, you can check out our list of vegan products. At least with this list someone has done the legwork for you. http://www.vegfamily.com/product-reviews/index.htm.
Not an exhaustive list, but we add to it monthly.
Once you get used to eating vegan it's not too hard.
Whey and casein are the two main dairy ingredients that find their way into a lot of stuff. Especially whey. It seems to be in anything that is in powdery form (cake mixes, bread mixes, bread, ice cream...etc.)
I read somewhere that there's a law that prohibits manufacturers from dumping whey into the sewer system as it's 2000 time more toxic than raw sewage. Of course, that didn't stop the companies from putting it in food. "Oh, can't put it in the sewer? What are we gonna do with it? Hey, I know, let's turn it into powder and sell it as cheap filler to other companies. Heck, let's give it to them for free, otherwise we have to pay high fees to have hazmat take it away..." Brilliant marketing.
I digressed... it's late. Off to bed.