Your local comic book store is the best place to get each issue, but if that's not an option, you can order it directly from the publisher, Boom Studios — but do note that shipping for one comic costs as much as the comic book itself!
Boom Studios also offers a "variant" cover edition of Peanuts #1, featuring a vintage 1950 Schulz drawing of Charlie Brown on a yellow background. But quantities of the variant cover edition are deliberately very limited, so your comic book store may not have one — and even if they do, it likely will be quite a bit more expensive than the regular cover version. For example, Boom Studios lists the variant cover edition on its website at a cost of $19.99! We think most folks will be happy with the regular cover version and don't need to spend money on deliberately scarce "collectible" versions of the comic.
Although this new Peanuts series is advertised as being monthly, that's not entirely true; the comic books will be published as a series of four issues over four months, followed by a break of a few months, and then another set of four, and so on. (So, Peanuts #2-#4 will follow from February through April, followed by a pause.) After each set of four comics is published, they'll subsequently be collected into a trade paperback edition. The first trade paperback, which will collect Peanuts #1-4 and presumably the special preview issue Peanuts #0, is due out in August and can be pre-ordered from Amazon. We've no information at this time regarding the possibility of bonus material in the trade editions.
So, if you want to experience the pleasure of getting a new Peanuts comic each month, now's the time to head out and get Peanuts #1!