How can my story be improved?

Very interesting story you have going there, but zombies are just incredibly stereotypical. I too am writing my own apocalyptic-based novel, it is about how water is becoming a rare resource in the world, and how eventually humanity destroyed itself over it, in the form of nuclear destruction. What you have written here defiantly caught my interest after reading the 1st paragraph. The only thing I suggest is you don't refer to them as zombies. It feels incredibly stereotypical, and you should rethink what these monsters are exactly. Create something new, like for example in I Am Legend which also takes place in NY, the monsters aren't really zombies although they act like it. They can't go out in the sunlight for long, they sleep during the day, they look less like green corpses and more like humans without skin. And they have some degree of intelligence as seen in the movie when they have mutated dogs that hunt alongside them. In other words, make the monsters in your novel more interesting. Describe them and give them special weaknesses and setbacks as your character discovers them in your novel. The amount of things you can put into the story is unlimited. There is no boundary that stops you from getting creative. Just think about this as you write on with your book. Best of wishes ;)