Hi Bas, if you are indeed interested in Tibetan Buddhism then I would recommend visiting a temple in Manhattan named Yeshe Nyingpo. The head Lama, Shenphen Dawa Rinpoche, is the son and lineage heir of his father, HH Dudjom Rinpoche who was actually one of the Dalai Lama's own teachers (he taught him Dzogchen, the highest practice of Tibetan Buddhism, as did Dilgo Khyentse Rinpoche as well). This is a good time because Rinpoche usually resides at his retreat center in upstate NY, but he is here in the city for the Tibetan New Year practices. What I would recommend if you had the time and resources, is to attend a teaching with Rinpoche-- they do not occur often but he is giving one after the Tibetan New Year on the weekend of the 6th 7th and 8th of March, beginning on the evening of the 6th. This is a teaching on the practice of Phowa. This is not necessarily a basic teaching, however it is one everyone should know, as it pertains to the most important time of one's life: the time of dying, but it also relates to the entirety of Dharma practice. SD RInpoche speaks perfect English, which is rare for a Tibetan Lama of his generation and experience. Because of his qualities and accessibility, I think this is the best place to go personally. However, the pujas such as the tsoks are performed very traditionally in the Tibetan language so it may be overwhelming and incomprehensible to attend one of these as a first approach. That is why I recommend attending a teaching with Rinpoche as an introduction. Here is a bit about him: http://www.tersar.org/?page_id=439
There are also classes with senior disciples regularly (weekly or bi-weekly) on Shamatha meditation, ngondro (tantric preliminary) practice, and practice sessions like Chod.
While some of the other centers Kirtu mentioned above are good places to explore, I will mention a few things regarding each one:
At Shambhala NYC you will mostly find teachings with the American senior students of Chogyam Trungpa Rinpoche who went on to follow his son, Sakyong Mipham Rinpoche. Sakyong Mipham Rinpoche does not visit often, perhaps once a year I believe. They have weekly Dharma gatherings (last time I checked) which give meditation instruction for free, a type of Shamatha (calm-abiding) meditation. This is a good introduction to a basic and quintessential meditation discipline and a foundation for everything else. That said, if you were wanting to jump into doing Vajrayana preliminaries or practices they are extremely slow and conservative about this and you would probably have to make your way through various programs for years before accessing those. This may be the perfect approach for some people, and may not be for others.
Tibet House has some visiting meditation instructors from various Buddhist traditions on a semi-regular basis.. usually local western teachers such as Sharon Salzberg. They also occasionally host seminars or teachings with visiting Tibetan Lamas or scholars, etc. including with Robert Thurman, the scholar who founded the center. It is a good resource, but not quite a living community at this point. It is primarily intended as a hub to assist in preserving Tibetan culture in exile.
The Palyul Center is a nice small temple in Chinatown, with a wonderful lineage mainly devoted to the late Penor Rinpoche. There is not really a resident Lama, although there is a Khenpo who teaches on occasion. They have regular practices, but with only occasional instruction. There is a new short teaching series on the Longchen Nyingthig practice of Chod coming up soon, which may be a bit complex for a beginner but you may have a deep karmic connection, in which case it would seem familiar, not complex. Overall it is a good community.
Khenpo Pema Wangdak is a wonderful teacher. He is a good resource especially if you feel a connection with the Sakya lineage, but he also teaches Tibetan language (I think he still does this) and the center is indeed small but well set up and cozy. I am not sure of the schedule but it is worth checking out.
There is also a Nechung center where a monk named Lama Pema Dorje leads weekly practice sessions or teachings. Often they simply watch video of the Dalai Lama's teachings and Lama Pema will elaborate a bit after. He is very devoted to HH the Dalai Lama. They mostly will recite prayers according to the Nechung Monastery's daily prayer book with a minimum of time in meditation. This may or may not be a good introduction depending on your preference.
I have no experience with the Shantideva center, so can not comment.
It could be a good thing to visit any or all of these places or teachers to see what you feel a connection with, energetically or aesthetically, or even program-wise.