No, you will not become a moderator immediately. If the site makes it out if private beta, there will be a thread on the per-site meta, accepting nominations for moderators pro-tempore. You can nominate yourself by adding an answer to that thread, and, I assume, the top-voted candidates will become that site's moderator team until graduation, when there is an official election, at sitename.stackexchange.com/election.
Read more here, the SE blog post about pro-tem mods.
I took part in the electing of the the pro-tems for the AI site, so here's a bit more in-depth explanation of the process:
First, there will be a Meta thread, asking for nominations, such as this one.
You can nominate other people, who then have to edit the nomination stating whether or not they accept the nomination, or you can volunteer yourself.
People will be able to vote on these nominations, if they happen across the thread on meta, or see it because it's featured.
After a while, the SE staff with choose a number of the candidates, usually three, to be the moderator team for that site. They contact the nominee via the email in their profile, verify things, etc. I don't know whether or not the mods are chosen only from those answers on that thread or not.
Then will come a thread on meta announcing the new moderators, who are then the moderators until the site either graduates, when there will be a real election which takes place at sitename.stackexchange.com/election, or the site closes.