Neither textbook is wrong, and it's important to learn both forms, but I would put my emphasis on studying the 요 form first, since you will be using this one more. Of course, the 니다 form is much simpler and doesn't take as much time to learn (making it also seem like a logical choice to study first so you can get going), but, hey, get the harder, more practical part out of the way first. Just as I would recommend an English learner study colloquial English before learning to talk like a news anchor (similar but not identical concept).
Yes, 요 form is informal but still polite, making it a safe bet for a learner to use when speaking or writing. 니다 form is both formal and polite, but in most situations you'll run into (especially early on), this will be waaay too formal. You'll sound like such a noob :)
Anyway, I would worry less about the names of things and more about the uses. Know when to use 요 form through examples. And likewise with 니다.
요 - 가요
니다 - 갑니다
요 - 해요
니다 - 합니다
요 - 봐요
니다 - 봅니다
Between each set, the meaning is exactly identical. The only difference is who you're talking TO (as opposed to about).
Now, I'm not sure how much of an explanation on conjugating you're looking for...feel free to comment.