The new movie Into The Spider-Verse looks absolutely amazing, and the comic that shares the same name make one of the best intros into comics you could want. The original comic Spider-Verse introduces to a lot of characters, writers have been collected in one big book.
This book is big, and a little pricey, but it includes the full Spider-Verse series, including every single related issue, so it's a pretty great deal. You don't need any other books to enjoy this really fun series. The only downside is that they've included the issues grouped by series, not by when they happened. If you prefer to jump around and read them in chronological order, How To Love Comics - Spider-Verse has a good issue-by-issue reading list.
For anyone who wants a little more lead-in, I've included some lead-ins below, as well as some ideas for series after Spider-Verse. The three books below are all set before Spider-Verse begins, but are not necessary to enjoy the story.
Amazing Spider-Man Volume 1 - The Parker Luck - This has no connection to Spider-Verse, or almost none. I include it because it's just one book before things get started, and it's a really clean starting point for Spider-Man, so some readers may be interested.
Amazing Spider-Man Volume 2 - Spider-Verse Prelude - This continues the Peter Parker storyline up to the Spider-Verse event. This also isn't necessary, but adds some nice backstory.
- Edge of Spider-Verse - This is a really fun book. As Spider-Verse was getting ready to feature many, many, many Spider-Characters, they decided to do a brief series introducing different alternate-universe Spiders. One of them, Spider-Gwen, was so wildly popular that plans for her own series began before her first issue of _Edge of the Spider-Verse) was even on shelves! If you only grab one other book besides Spider-Verse, this should be it.
I have a few articles that have starting points after Spider-Verse, including Spider-Woman and Spider-Gwen. In addition, the series below, Web Warriors of the Spider-Verse, continues directly after the events of Spider-Verse with many of the same characters and settings:
A few years later, another event, Secret-Wars included a tie-in very confusingly also called Spider-Verse, which I'm sure has confused a lot of new readers. It's a fun tie-in to that event, but it is not the original Spider-Verse story, so be careful not to get them mixed up.