So I publish my first foundry mission today (or so the Foundry TOLD me). Then I logged out, logged into my primary character, get an email that says "your mission successfully published", and open up the community authored tab, all excited, and it's not there. Not by searching my author name, not by the mission name, nothing. What gives?

It's in the Review Content tab. It needs 5 plays, with ratings before it shows in the regular community authored listings.

As Kirksplat says, it will be invisible to the general public until it gets 5 reviews from designated reviewers. I'll just add that its easy to become a designated reviewer, just go to the community authored tab and click the button that says "Become a reviewer"

Also my other standard answer to the question: make sure you're on a character that is high enough level to see the mission. Mission level is based on what type of enemy NPC groups you use. Klingons are any level, Romulans are 16+, etc.