Do you know of a good place to get renters insurance through?
Renters insurance is certainly a good idea. (Although New York City law offers many protections to tenants.)
Here are some of the basic reasons to get renters insurance. Mostly, it's to protect you against property damage. The owner of your apartment usually has insurance on the building but you are responsible for whatever is in your apartment and whatever damage you inflict on it (or whatever damage is inflicted on people in your apartment, who can sue you if they are injured in some way).
The good news is that it's not that expensive. We calculated what it would cost to insure a small 1 bedroom apartment in Manhattan with standard household items with State Farm and it works out to about $200 a year (or $16 a month, and some change). (The State Farm page also has some good information about why you might need renter's insurance, too, and what your liabilities are.)
We feel reluctant to recommend a specific company, given our own minimal experience with insurance (although the above paragraph does look like an ad for State Farm; to be fair, other insurance companies: Geico, AAA, All State), so we'll suggest instead that you just do your research. Get a couple of quotes, read about different policies, ask your neighbors, go with a reputable company.