Welcome to the board.
If you are having problems with tears in your NY style dough, you may want to consider the following tips based on my own experience and the experience of others on this forum who have had good results with the NY style.
1) Find a good NY style dough recipe--one using high-gluten flour and a relatively high percent of water relative to the weight of flour. This ratio is called the hydration percent and should have a value of about 58-65% for a NY style dough. My personal preference is 63%. The combination of using a high-gluten flour and a high hydration percent will alone help minimize tears in the dough. It will help if you have a scale to weight the flour and water especially, since if these weights are off too much the results may not be what you'd like and will in many cases require you to make adjustments to flour and/or water at a later stage. The adjustments, if not properly made, can themselves throw off the hydration percent.
2) Knead the dough sufficiently, i.e., don't underknead the dough since this can lead to tearing also when you are ready to work with the dough to make pizzas. When in proper form, the kneaded dough should be soft, smooth and elastic with no tears on the outer surface, and not overly dry or wet, but rather tacky. If the dough is overly sticky, knead the dough by hand for a minute or two. In most cases, the stickiness will disappear as the flour absorbs the water. Only if the stickiness persists should you add more flour, and just the absolute minimum to overcome the stickiness. Keep in mind also that if kneading is being done completely by hand, it will take longer and be more difficult to knead a dough made with high-gluten flour to get the finished dough ball to the desired form. But it can be done.
3) Add some oil, such as a light olive oil, to the dough. This is best done as a separate step after the water has been added to the rest of the ingredients and kneaded into the dough. The oil helps produce a dough that can be more easily stretched. Usually the more oil that is added, the easier it will be to stretch the dough (the oil coats the gluten in the dough and makes the dough more extensible, or stretchy). But don't go overboard with the oil, since not much is required for a NY style dough.
4) As soon as the dough has complied with the preceding steps, put it in a suitable container (plastic storage bags and metal containers are good choices) and refrigerate, for 24 hours, if possible. That will give the gluten in the dough plenty of time to relax and make the dough easier to handle and it will also produce flavors of fermentation that will improve the taste of the crust. If you want to extend the useful life of the dough from the outset, say, beyond 24 hours, you can add a bit of sugar to the dough and/or use cooler water when you make the dough to begin with. If you can, it is usually a good idea to temperature adjust the water you use to achieve a finished dough temperature as it comes out of the machine (or out of your hands if hand kneading) of around 80-85 degrees F. This can be done fairly easily by using cooler water in summer than in winter. The objective is to prevent the dough from overrising in the refrigerator because of excessive heat, which speeds up the rate of fermentation beyond what is desirable.
5) Once you are ready to use the dough and have removed it from the refrigerator, let it set at room temperature, lightly covered with a sheet of plastic film, for about 1-2 hours, or until the dough temperature is around 55-60 degrees F. The actual time will depend on the temperature of the dough when it comes out of the refrigerator and the temperature of the room where you will be making the pizza. All other things being equal, the dough will reach the above temperature faster in summer than in winter. Once the dough reaches the proper temperature, it can be safely used thereafter for about another 1-3 hours (depending on room temperature).
6) If you have succeeded with steps 1) through 5), you are unlikely to have any problems with your dough. To shape the dough, put a small amount of flour on your work surface, carefully flatten the dough ball into the shape of a disk (you don't want to force out the gasses trapped within), and lightly coat your dough ball on both sides with flour. Then press down on the dough with your fingers, pushing the dough outwardly gradually from the center to the edges to enlarge its diameter. When the dough reaches a size where you can pick it up and shape and stretch it, place your knuckles under the dough on opposite sides and start stretching the dough outwardly while at the same time rotating the dough a bit at a time ("flicking" may be a better term) so that the dough doesn't become lopsided. Try to avoid working the middle of the dough too much since this can lead to thin spots and, possibly, tearing, because of the weak spots. Work mostly the outside of the dough. When the dough reaches the desired final size, place the dough on a lightly floured peel (or use corn meal if you'd prefer) or onto a seasoned pizza screen. Dress and bake.
Feel free to ask any questions.