I worked in NY and CA in 2023. My domicile is NJ. How many state taxes should I file?

Yes.  You will file a resident NJ return and assuming you did not establish residency in NY or CA during the year but you meet the requirements to file a return in each of those non-resident states (see filing requirements below), you will file non-resident returns for both NY and CA and allocate the respective part of your income earned in each those states (steps below).

Here are the rules for NY residency and filing requirements and for CA residency and filing requirements for tax purposes.

NJ, as your resident state, taxes income no matter where it is earned and the non-resident states only tax the income sourced to their states.  That sets up double taxation on the income.  To mitigate that double taxation the resident state (NJ) offers a credit for taxes paid to other states.  You should complete the non-resident state interviews prior to starting the resident state interview in TurboTax.  In that way, the figures needed to calculate the credit on your NJ return will be available for TurboTax to automatically use.

1. In the "My Info" section
2. Edit your personal information
3. Indicate you lived in NJ on Dec 31st
4. Assuming you did not establish residency in any other state during the tax year answer "NO" you did not live in any other states.
5. Continue
6. Still in the "My Info" section
7. Scroll down to "Other State Income"
8. Answer "Yes" you earned income in another state
9. Select NY and CA 
10. This will generate the non-resident interviews for those states.