Check out the Ladybug family of magazines (I believe it is Carus Publishing.)
Babybug is the monthly publication for the littlest ones. Many public libraries with children's rooms have a subscription; I think it's a high quality magazine. There's no advertising! They do include a post card mailer between the pages offering subscriptions but it pulls out.
Babybug is engaging & hardy (the pages hold up, though historically they've been a bit tougher, it's still good-quality) and very child-friendly. The format is stable, with one story ("Kim & Carrots") featured in every issue (always a different adventure for preschooler Kim), and changing features that follow a predictable structure (rhymes & bounces, Mother Goose verses, finger play & tickle play for babies/parents) and generally reflect the seasons & sometimes holidays.
It doesn't progress with age but the publisher has other mag. options for as they grow (Ladybug, Spider.)
at 12:59 PM on Apr. 1, 2013