Hey Samantha! Best of luck on your vegan journey :) I'm a vegetarian but I eat a vegan diet for a significant portion of the year, so I hope I can be of some help.
In general, most things that have more than one ingredient listed are processed. You don't necessarily have to avoid all processed foods (but more power to you if you do!); in general, it's best to stick to whole grains, legumes, and fresh produce.
For grains, buy some brown rice, whole wheat pasta, quinoa (also a great protein source!), barley, etc. Technically, they may have been "processed" since they're in a package, but any processing would be minimal, so they would be a good choice. They are a good, healthy staple to keep on hand and they will allow you to prepare a wide variety of foods.
Legumes (black beans, lentils, etc.) are a great vegan source of protein. If you buy them in a can, they would be considered processed. However, if you rinse them to rid them of any excess salt, they would still be a good choice. For something slightly less processed, you could buy them in a bag and prepare them yourself.
Fresh produce should be pretty self-explanatory. There's no ingredient list needed - you know it's just one ingredient! You really can't go wrong with the fresh, unpackaged stuff.
One thing that I like to do is prepare a big batch of grains in advance - quinoa or rice, usually. Then I can use them throughout the week, pairing them with different ingredients for completely different meals: black bean quinoa burgers one day; quinoa with sauteed tomato, onion, spinach, and mushrooms the next.
Remember: the closer a food is to its natural state, the less processed it is. Not all processed foods are equally bad: obviously, a can of black beans is much better for you than a Twinkie! You can't really go wrong with fresh fruits & veggies, whole grains, and legumes.
Hopefully that helped somewhat! :)