Pour the warm water in a large mixing bowl. Add the sugar and package of yeast. Stir the mixture slowly until yeast and sugar are dissolved. Let sit to allow the mixture to "mature" for about ten minutes or so. The mixture will begin to react; clouding and forming a foamy "head" on the surface of the mixture. Add the salt, olive oil and vegetable oil and stir again to combine and dissolve the ingredients. Add 1 cup of flour and whisk in until dissolved. Add the second cup of flour and whisk it in. Add the 3rd cup of flour and combine. By now the dough mixture should be fairly thick. Add the last cup of flour and, with your hands, begin to combine and knead the dough.
Remove the dough ball to a tabletop to knead it if the mixing bowl is not large enough to knead in. You may need to add a dusting of flour from time to time to reduce the stickiness of the dough. Be patient, folding the dough ball in half and then quarters, over and over again for about 8 to 10 minutes, (or about 100 "cycles".) Kneading by hand is laborious, but very important. Better to over-knead than under-knead. You'll know you've done well when the ball no longer sticks to your hands. It will become a smoothly-textured ball slightly larger than a large grapefruit. Coat the dough ball with a thin layer of vegetable oil, and place it in the bottom of a large mixing bowl which has also been coated on the inside with vegetable oil. Stretch a piece of kitchen film over the top of the bowl and set it in a warm place such an as un-lit oven, (ambient temperature of 70°F to 80°F). Allow the dough to rise, undisturbed, for 60 to 75 minutes. The dough will have grown to at least twice its original size. Take 1/3 of the dough and make it into a ball while working any air that could be trapped inside out of the ball.
If you wish to store the dough, by either freezing or refrigeration, you can place the dough balls in zip-lock bags. Squirt a little vegetable oil into each of the bags to keep the balls moist and pliable and to ease removal when ready for use. If you choose to freeze or refrigerate, the dough balls may continue to rise until they are substantially cooled down or frozen, which is OK as long as they don't break out of their bags. If they do, mold them back down into balls and re-bag them.
If you choose to continue with making the pizzas now, or once thawed from being frozen, here's what to do. Take one dough ball and stretch and push it out using your fingers to fill a 12" pizza pan. Try to make the dough so it will be about the same thickness around the entire pan. If needed you can push dough from an area that has too much dough to an area that is thin from not enough dough. Leave the panned dough to rise for about 20 minutes to allow it the opportunity to rise a little before preparing the pizza and cooking.
Now that the 20 minutes of rising is complete, take a fork and poke the entire panned dough with the fork to release any pockets of air and to allow the sauce a way to hold onto the dough. Spoon pizza sauce onto the dough starting from the center working your way out to the edge. Cover the pizza sauce with a layer of mozzarella cheese. Add any toppings that you would like on the pizza. If adding meat and vegetables as toppings it is recommended to start with the meat first and finish with the vegetables. Sprinkle a light layer of mozzarella cheese on the top of the pizza to hold the toppings in place. Cook in a preheated oven at 450° for 8 to 12 minutes or until the cheese is melted and the dough is cooked.