I am on a F-1 VISA. I am technically a resident of NY and I did an internship in OH

You are a full-time resident of New York if you lived there for the whole year, including the time you were in Ohio, since that was just a temporary assignment. To find the taxes paid to another state, you can look on the state tax summary for Ohio in TurboTax as follows:

While working in the Ohio state return:

1. Click on Tax Tools in the left menu bar
2. Click on Tools
3. Look under Other Helpful links….
4. Choose View Tax Summary

Thank you so much!

Since I'm receiving a refund from the state of Ohio, where I paid state taxes, does this still mean I've paid taxes to another state, right? Would I have credit for taxes paid in OH?

And if so, for the "taxes paid to another state" section on my NY state tax form (in the "Credits and other taxes: section), they ask for the following information:

- Tax Computed on Your Ohio Return

- Ohio tax paid by you/on your behalf by an entity

I am not sure where to find this information. Would it be on my W-2? Or should I leave that empty?

Thank you so much in advance!

If you are only filing an OH return to get a full refund of withheld taxes, there will be no credit on the NY return, because you did not end up actually paying any OH tax. 

If you had Ohio city income tax (box 19 of your w-2) withheld, you cannot get that refunded.  But NY will give you a credit for that.

Thank you! Do you happen to know in what portion of the NY State tax form on TurboTax I can include that local (Ohio City) income tax so they give me credit for it?

I'm not specifically familiar with the NY software. It's usually on a page titled "Credits and  Taxes" and a line "Tax paid to another state". Local taxes would be a sub section or separate line