The first step to seeing your designs in print is to find is a call for submissions, in which the editor will tell you what kinds of designs they're looking for (and how and when to send in your ideas). Here are some links to where submission calls are posted:
Interweave: knitting magazine calls, crochet magazine calls, and spinning magazine calls.
Knitty: submission guide.
Twist Collective: submission information.
Knotions: submission guide.
Knit Picks: IDP and submission calls.
PomPom Quarterly: submission information.
Holla Knits: submission calls.
Knittin' Little: submission calls.
And the best one-stop-shop of submission calls, the Designers group on Ravelry. Various magazines, yarn companies and such post here regularly, and sometimes unique one-time calls pop up as well.
While this is by no means a complete list, it's enough to see that there are a lot of options! How are you ever supposed to choose where to submit your designs? Short answer: choose a publication that wants what you do best. Some publications focus on fashionable clothes, some on accessories, some on kids items, and others publish a wide variety of patterns. So read the submission calls thoroughly, and do a little research (looking at the magazine's website, or checking out the patterns published in previous issues on Ravelry) to see if the magazine's aesthetic fits with your designs.
On the other hand, maybe you know what publication you want to design for but you aren't sure what to design. My advice is to only submit ideas you are confident you can pull off. Never knit a sock before? Don't submit a sock idea. Uncomfortable with grading? Go for a pattern that's only one size. Be sure to propose something for which you are familiar with the stitches, fit and construction. Even better, make a full-sized sample beforehand to accustom yourself to the design and pin-point any problem areas. It is worth taking the extra time to flesh out your idea beforehand to help prevent mishaps if it is accepted.
So, prospective designers: check out some of the links above, find a publication you'd like to design for, and start thinking of pattern ideas. Then meet me back here next week, where I'll give some advice on the actual proposal.