I've just moved to California but will continue to do work for a Florida company. How does this work tax wise?

If you live in CA, then during that period your worldwide income is taxable by CA.
Your U.S. Tax Return: Working or Living in Two or More states

Make sure that your employer deducts CA income taxes from your paycheck. If they can't (and given that FL has no income tax I wouldn't be surprised if this is beyond them) then you should file quarterly estimated tax payments with the CA Franchise Tax Board. Otherwise if you wait until next April to pay your income taxes to CA you could end up owing interest on the underpayment.