Your question is a good one. Not everyone has all the fancy equipment that experienced home pizza makers usually end up accumulating with time. Most of us started simply and worked our way up as we gained experience and had more money to devote to the hobby. I believe it was Confucius who said that "A journey of a thousand miles begins with a single step."
Without equipment, like a stand mixer, food processor or even a bread machine, you are pretty much relegated to hand kneading. All-purpose and bread flours can be kneaded that way but it gets harder to do with high-gluten flours, which you are unlikely to find in any event in the food stores where you regularly shop. With a suitable dough recipe, you should be able to make a NY style pizza with a thin crust using a good bread flour. The best bread flour from my experience is the King Arthur brand. It is sold in some supermarkets and specialty food stores, but if you can't find it where you shop there are other brands, like Pillsbury and Gold Medal, that should work for your purposes and budget.
For baking the pizza, you should perhaps invest in a pizza screen. They come in all sizes. If you want to make the largest size that your oven is likely to handle, then I would go with the 16-inch size. Otherwise, I would look at 12- or 14-inch screens. Or you can get several screens of different sizes. With a pizza screen, you don't need a pizza stone. If you don't know what a screen looks like, go to Reply #130 at http://forum.pizzamaking.com/index.php?topic=389.msg4545#msg4545
. Screens are very inexpensive--typically only a few dollars apiece. You might be able to find them at a local restaurant supply store that carries pizza items, or you can go online to any number of places to buy them. If you need a link or two to places that sell them online, let us know.
In lieu of using pizza screens, you can also consider using tiles as a much lower cost alternative to a pizza stone. Tiles are very inexpensive and are widely available at places like Home Depot or Lowes. You want the red unglazed quarry tiles that are safe for baking. If you go the tiles route you may also want to invest in a pizza peel (if you don't already have one) to get your pizza onto and off of the tiles. The peel also comes in handy in removing pizzas baked on a screen from the oven. I would look for a wooden peel for now. The places that sell pizza screens also usually carry pizza peels. They usually sell for less than $20.