Vermont will consider you a resident for tax purposes as soon as you establish your domicile (your permanent residence) in Vermont. Therefore, if you have literally moved to VT (as opposed to being there temporarily), you are a Vermont resident as of the date you began living in VT - even if you didn't yet have a utility bill or your Vermont D.L. http://tax.vermont.gov/individuals/income-tax-returns/who-needs-to-file/resident
If you moved during 2017, you will have to file a 2017 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
The income you earn in NY is taxable by NY whether you are a resident or a non-resident. Therefore it is OK for your employer to continue withholding NY income tax, even after you become a VT resident.
In the future, if you continue to live in VT and work in NY, you'll be filing a non-resident return in NY and a resident return in VT. VT will credit you for taxes paid to NY, so you won't be double-taxed.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
