Should I set resident state NY where my college at?

Your resident state is where your permanent home is. For most college students, that's the parents' home. Being away at college is just a temporary absence.

If you are an Illinois resident, but you earned money in California and New York, you will have to file nonresident tax returns in CA and NY, as well as a resident tax return in IL.

Whether your parents claim you as a dependent has no effect on where your residence is. Note that the question you are asked on the tax return is not whether someone else claims you as a dependent. It's whether your parents or anyone else CAN claim you as a dependent. If they can claim you but choose not to, you still have to answer Yes to that question. Most college students can be claimed by their parents.