I lived in CA for 3 months, then moved to NY (due to work for 3 months), then moved to PA (where I currently reside for work). How to make PA my "resident state"?

In the My Info or Personal Info section of TurboTax, indicate that you lived in PA.  Then, in the State section, you'll indicate that you are filing a Part-Year Resident return, because you were not a resident of PA the entire year.

Depending on how much money you made in 2019 while living in California, you may have to file a Part-Year Resident tax return there.  See this CA FTB website to know if you are required to file a Part-Year Resident tax return for CA. Even if you are not required to file a CA tax return, you may want to do so if you will receive a refund of CA withholding.

If you did not intend to live permanently in NY, you file a nonresident return if required.  See this New York State website for the filing requirements.