You will need to file a nonresident income tax return for Michigan to claim a refund of the Michigan tax withheld.
Please see the link below for more details on the forms required to be filed as it will differ based upon your particular tax situation. For your particular set of circumstances, look at the bottom of page 1 on the link.
Michigan filing requirements
You will also need to contact your employer to submit an Employee's Michigan Withholding Exemption Certificate as shown in the link below.
Michigan withholding exemption form
Since Michigan and Ohio have state reciprocity, you should only have income taxes withheld in your resident state of Ohio.
What is a state reciprocal agreement
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