I’ve covered this a lot in previous Asks, but in short I think relevant experience always helps and can never hurt. However, what’s relevant doesn’t always need to be a internship at one of the big six publishers, but if you’re in a position where you can apply for such internships (enrolled in college, living in or around New York City), it’s a very good idea to attempt to get one.
Each job in publishing (and there are many different kinds) comes with different job requirements, so while I don’t want to generalize, a demonstrable love of and knowledge of books is very important. Organization and office experience also a plus, and you can get that anywhere. Also consider working at a bookstore, or at a small press or university press, or a literary magazine, or consider a publishing course (Columbia and University of Denver have great programs). All of that will give you relevant experience.
There isn’t a one-size-fits-all path to publishing, just like there’s not a one-size-fits-all path to getting published, but those are all great ways to start.
All of the imprints under the Penguin Young Readers Group umbrella have their own submissions guidelines, which you can find on each imprint’s profile page on Penguin.com. Go here and mouse over “Young Readers Division” to choose an imprint. Anything submitted directly to Penguin by an author (except in cases where the editor requests it from an author that they’ve known or met) is considered an unsolicited submission; for the most part, it’s a much better idea to get an agent first who can submit your manuscript directly to an editor at Penguin, which would be a solicited manuscript. Hope that helps!