Personally, I think the best option is to learn a few of the major ingredients that are common: whey, gelatin, albumen, Casein, lanolin, etc.
But memorizing the whole list is nearly impossible. So I suggest looking for "vegan" on the label and if you don't find that then quickly scan the ingredients and choose items that have very few ingredients that are easy to read and understand. That will elliminate a lot of items from the grocery store, but that's OK. Focus your meals around things like beans, rice, veggies, fruits, other grains and legumes.
Just remember not to fuss over it too much. I'm not saying ignore gelatin or anything, but if you're in the store and you see a cereal or something that contains a vitamin which you read on the internet is sometimes obtained from a chemical which is sometimes obtained from a chemical which is sometimes obtained from an animal product, then I wouldn't worry about it. Unless, of course, you are in a situation where you can afford to worry about it, but most people aren't.
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