I only mentioned owner reposession because I have had one lease where it was specified that if the owner wanted to move in, she could give 30 days instead of riding out the full-year lease term... but it was specified in the lease at time of signing. So if you didn't provide, probably not an option for you. (In retrospect, I'm not even sure that was necessarily legal... but it wasn't an issue.)
But you answered your own question about notice time. You quoted the line stating that the tenancy does revert to month-to-month (tenant-at-will) and you just need to send out the 30 day notice. How that notice will need to be presented will depend on your local laws (sorry, I'm from MA--can't help you there). It varies from door-nailing to certified return receipt or constable service... maybe another forum member can guide you?
But know up front that day 31, if they're still in there you can't just change the locks and start moving stuff... Day 31 you have to begin proceedings for a court-eviction. If they're stubborn or know the laws better than you, they can delay the proceedings for quite some time.
Run out to the bookstore and grab a local landlord/tenant law book today!
Good luck!