Dialects are region to region, comune to comune. There are tons. Its my understanding that every one is taught true Italian nowadays. So most people should understand you from any Italian program you try to learn from. Its the older folks speaking in dialect you might not have luck with.
Standard Italian taught in the schools, and spoken in Italy, is based on the Tuscan dialect and was adopted in Italy after its unification. Before this point, I believe this dialect was only used by the upper class in Italy.
There are Italian courses taught at various branches of the City University of New York, so you might want to look into that possibility. There are also private schools throughout NYC where Italian language instruction is given. Below are some links you might find helpful.
I found this information which you may find interesting:
Many of the Italian dialects are now disappearing. The younger generations are not learning them anymore or they are using them strictly with their older relatives and most of the dialects were never recognized or promoted by the Italian government. There are a few exceptions, and local languages that are considered co-official with the Italian, like the numerous Sardinian languages for example. In many cases though, even when the dialects are lost, it is still easy to identify an Italian by the way he speaks. The reason is that every local dialect influenced the way people learned Italian, and thus the "accents" were born. Italian accents are slight variations of the Italian language, variations in prosody and rhythm (the way a language "sounds"), as well as in grammatical structures and lexicon. In some cases the same apparently Italian word can have two completely different meanings in different regions. For example "guanciale" in Tuscany means pillow, while in Lazio it's the pork cheek. In other cases an Italian word is changed so much that it becomes unrecognizable, like the Italian "andiamo" can become "namo" in Rome. Some other times the accents take words directly from the dialect that don't even exist in the official Italian.