"While I'm not planning on staying here permanently..."
If you're in NY temporarily in order to attend school, and you intend to return to your home in Wisconsin, then for tax purposes you remain a resident of Wisconsin. In tax terminology, WI would still be your domicile state, and ALL your income would be taxable by WI. Any income you would earn by working in NY would also be taxable by NY. In this scenario you would file both a non-resident NY tax return reporting your NY income and a resident WI return reporting all your income, including that from NY.
But if you've abandoned your WI domicile and do not intend to return, and you've made NY your new main, primary home, then you became a domicile resident of NY (and ceased being a resident of WI) when you began living in your new NY home. In this scenario you would file as a part-year resident of each of the two states. See What is domicile? on page 3 of this reference for further explanation:
https://www.revenue.wi.gov/TaxForms2017through2019/2019-Form1NPR-inst.pdf#page=61
Finally, even if you intend to return to WI, if you "maintain a permanent place of abode in New York State for substantially all of the taxable year and spend 184 days or more in New York State during the taxable year, whether or not you are domiciled in New York State for any portion of the taxable year", NY will consider you a NY resident for tax purposes - making all your income taxable by NY. An example of a "permanent place of abode" would be a rented apartment. A college dorm is not a permanent place of abode.
https://www.tax.ny.gov/pit/file/pit_definitions.htm
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
