New York City
I assume you mean Spring of 2015. That said, there's still more information needed before any of us could offer an answer. On tghe other hand, if you're looking for neighbohoods seved by public transit, the good news is that all of them are! Some have better connections for particular destinations than others, but one fare will take you anywhere in the city even if you need to make transfers.
How many of you are coming? Are children included? What are the things you're most interested in seeing or doing? What's your budget? Those are the things that will help make recommendations that work for you.
One thing you should be aware of in looking for apartments: this is a hot-button issue lately, as New York laws prohibit most rentals under 30 days, and you may read articles about lawsuits and the like. However, apartments that are in houses that are not classified as "multiple dwellings" are perfectly legal, and there are many here in Brooklyn and in other parts of the city, though few in Manhattan.