I don't know in what neighbourhood you are but if you are near the East Village in NY it is worth checking out the 14th Street Y (between 1st and 2nd Ave). They have mommy groups and exercise classes where you can bring your baby along. For me it was a good starting point to meet other moms and go from there. Also, it is one of the only places where you don't have to be a member to join the mommy group or the mommy&me classes. If you are on the West side, there is a center called Babymoves, it is on Perry Street. After summer they go back to their normal schedule and they have indoor open play hours where you can go with your baby. It costs $12 per session. I have also heard of Scandinavia House where you can go on Sat and Sun and pay $5 or something like that so use their indoor play space. There is also an organisation called Appleseeds where they have a huge indoor play space but you have to be member and it is pretty expensive.
Those are some starting points for you, good luck!