I moved to California from Arizona on Feb 1, 2017. Do I report my income for January in CA or AZ? How do I show part time residency in CA?

Your income for January should be reported to the State of Arizona, as you earned it there.  If you lived part of the year in California and part of the year in Arizona, you will file two Part-Year Resident returns, one for the time you lived in California and one for Arizona.  

In the "My Info" Personal Info Summary section of your Federal return, TurboTax asks if you lived in another state.  Be certain you have answered this question Yes.  Here's how to do it:

• In your open Federal return, choose My Info in the left bar
• Continue to the Your Personal Info Summary screen.
• Choose Edit next to your name and scroll down to section 2. Tell us the states you lived in.
• Enter your state of residence on December 31, 2017 (California) and select Yes to I lived in another state in 2017.
• Enter Arizona and the date you moved to California.  (This question is tricky.)

These entries should generate the two Part-Year Resident returns you need.