Do I simply serve a month-to-month tenant with a 3-day notice, and then file and eviction action, or do I need a 15 day termination of the rental agreement, and then take to small claims for the back rent?
You should hand-deliver a Pay or Quit Notice aka three day notice.
You should also mail certified mail a termination letter, which states that rent must be paid to cover all unpaid balances and the current month up until the date of termination.
Then, file a Count one and Count two lawsuit if they don't pay the unpaid rent.
If they do pay the unpaid rent, then you see if they leave January 1st.
If they don't , then you can either file a lawsuit for removal of holdover tenant or charge them double rent and give them a new Pay or Quit Notice once January rent is late.
One lawsuit is better than two.
Two is twice the money for filing fees and once the tenant leaves, you don't know where they are for purposes of serving them personally (which is mandatory for money damages lawsuit).