You will have to file NY, NYC, and MI tax returns. New York and New York City will tax all of your income as a statutory resident. MI will tax you on all of your income and provide a credit for the taxes paid to New York and New York city. This will reduce your MI tax liability to help avoid double taxation.
MI, NY, and NYC all tax residents on all income so it is required you report all of your income. Essentially, you will end up paying the highest tax rate but will be given a credit by MI for the taxes paid to NY and NYC. You could make sure your employer is also doing NY and NYC withholding on your behalf.
Michigan does offer other credits, but there are qualifying factors such as income level and taxable value of the home.
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