How do I report my earnings to each state if I worked remotely for a company based in the Los Angeles area all year, but moved from LA to New York City in March 2018?

[TomD8]

Your earnings during the time you were a CA resident are taxable by CA.  Your earnings after becoming a resident of NY are taxable only by NY, even though you were still working for the CA-based company. (CA does not tax non-resident remote workers.)  You became a NY resident (and ceased being a CA resident) on the day you began living in your new home in NY.

You'll be filing a part-year resident return in each of the two states.  This FAQ explains how to do that in TurboTax:  https://ttlc.intuit.com/questions/1901227-how-do-i-file-a-part-year-state-return