Kneading The Dough:
If you have a stand mixer, that will be great. Use a hook attachment, start at a slow speed and gradually increase the speed. Once you see the dough pulls away clean from the side of the mixing bowl, you know that you got all the measurements of ingredients right and the kneading process is done.
If you don't have a stand mixer, not to worry. You can get the same result from kneading with your hands. After combining all the ingredients with a spatula, set the spatula aside and get your hands into the mixture and start kneading inside the mixing bowl. Most of the time, a recipe will call for transferring the dough onto a floured working surface and start kneading. But, I find that kneading inside the bowl will ensure that all the crumbs of flour will be kneaded into the dough without adding additional flour.
To check whether the dough is kneaded properly, we do a "window-pane" test:
Pinch a small piece of the dough, flatten it out with the palm of your hand or a rolling pin to about 1/8 inch thick. Rise the flattened dough to a light or sunlight. If the dough is translucent and you are able to see the shadows of your fingers, the dough is kneaded properly. If not, continue kneading until the dough has passed the test. Knead in the small test piece back into the dough and form into a ball.
The Shaping Process:
It took me almost a year of practice to get the shape almost as round as a circle. Try not to use any rolling pin (unless you are making Roman Pizza), as the rolling pin will knock all the wonderful air out of the dough. We have the best tools in the world - our hands.
Dust working surface lightly with flour. Flatten the dough out with the palm of your hand. Lift the dough up and pinch 1/4 inch away from the edge of the dough with your thumbs and fingers as you swirl the dough in a circular motion. This will create that iconic pizza border.
Place the dough down on the working surface. Using your fingers, gently pull and stretch the dough in a circular motion, leaving that 1/4 inch border.
Continue pulling and stretching until the dough has become relatively larger in size.
Lift the dough up. Using your knuckles (like you are in a boxing match, LOL!), gently stretch the dough while swirling in a circular motion. At this stage, your dough is at its most vulnerable stage, which means, if you stretch it too much or too harshly, there will be holes. If there is/are holes, gently pinch up the holes. As soon as the middle is translucent, stop stretching.
So there, you have successfully shaped a pizza with your hands! All it takes is patience and practice. You can always form the dough back into a ball if you feel the shape isn't right or the holes are too big to be pinched and sealed. Let the dough ball relax for about 15 minutes, before start shaping again.