To answer your question about how to file:
If your domicile (your permanent home) is in NY, and you are in CT temporarily to attend school, you remain a NY resident for state tax purposes, and ALL your income is taxable by NY.
But if you worked in CT, your income from that job (or jobs) is also taxable to you by CT as a non-resident..
In that situation, you would have to file a non-resident return in CT AND a resident return in NY. You'd be able to take a credit on your NY return for taxes you paid to CT, so you wouldn't be double-taxed.
This TurboTax FAQ explains how to file a non-resident sate return: https://ttlc.intuit.com/questions/1901271-how-do-i-file-a-nonresident-state-return
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
