I only lived in New York for two months last year for an internship. Do I file as a part-year resident or a non-resident?

You file as a New York nonresident.  An internship (related with schooling) is considered a temporary absence that does not change the residency status of your original state.  Thus, if your resident state has an income tax, you will be taxed on your income there as well, but they will give you a credit for the income tax you must pay to NY on the same income.  Prepare the NY return first so TurboTax accurately calculates the credit.