Reddit mentions of Spider-Man: Season One

Marvel: Season One

http://www.amazon.com/Fantastic-Four-Season-Roberto-Aguirre-Sacasa/dp/0785156410

http://www.amazon.com/Spider-Man-Season-One-Cullen-Bunn/dp/0785158200

and a bunch more.

Marvel's been doing a series of graphic novels called "Season One." It's largely updated origin stories for different characters/groups, as well as a seemingly random recent issue in one hardcover. Thus far, they've released Fantastic Four, Daredevil, X-Men, Spider-Man, Ant-Man, and Hulk. Doctor Strange is scheduled for next month.

The two others in this thread gave some good suggestions, and I just want to throw a hardcover collection into the mix: Spider-Man Season One. I know the Season One series got a lot of flack for not being 100% true to the origins, and being slightly creative with the stories, but for a child the Season One series is really good if you ask me.

It gives the origin story of Spider-Man in a "modern" retelling along with his first major villain battle, it's 100+ pages or so and only a few bucks. https://smile.amazon.com/Spider-Man-Season-One-Cullen-Bunn/dp/0785158200?sa-no-redirect=1

Like I said, I wouldn't recommend it for a Spider-Man/Marvel purist, but for people who are new to comics or children I think it's great.

I would also recommend picking up collected editions/hardcovers of longer runs, but you have to be careful not to be overwhelming at the same time.

http://www.amazon.com/Spider-Man-Season-One-Cullen-Bunn/dp/0785158200

You should still give Amazing Fantasy #15 a read. Theres a million reprints of it out there, no reason not to read it as a Spider-Man fan.