What kind of dough and in what quantity will pretty much govern whether you can make the dough by hand. On the low end of the protein/gluten scale, from cake flour, 00 flour, pastry flour, and combinations of such flours, up to and including all-purpose flour, you should have no problem mixing and kneading by hand, even for fairly large amounts of dough. When you move up the next notch--to bread flour--the kneading will get a bit more difficult to do by hand, but you should still be able to do it with a little more effort, and especially so if you aren't planning to make a huge quantity of dough at one time. When you get to the upper end of the protein/gluten scale--to high-gluten flour such as King Arthur's Sir Lancelot flour or its equivalent--kneading gets more difficult and especially so if you plan to make a huge quantity of dough at one time. Since I have several machines at my disposal, I have not tried kneading a dough using the high-gluten flour entirely by hand. In fact, King Arthur says the Sir Lancelot flour should be kneaded only by a mixer, processor, or bread machine to develop the flour's gluten. Unfortunately, some of the best styles of pizza, such as the New York style, call for high-gluten flour. However, many of the recipes for New York style doughs (and others as well), often call for bread flour as an alternative to high-gluten flour, and in such case you should be able to knead the dough by hand. It will just take a little work. My best advice is to try making doughs using bread and high-gluten flours and see what results you get. What is key is getting the proper gluten development.
The two usual methods to knead dough by hand is to do it in a bowl or on a work surface. In the bowl approach, the dough ingredients are placed into a bowl (sometimes in a specific sequence) and, using a spoon or your hands, the dough is mixed and kneaded in a rotating (usually clockwise) manner while turning the bowl. This action in effect simulates the action of the dough hook of a stand mixer or the blade of a food processor (both of which operate much faster, of course, than your hands). The kneading continues until you get a dough ball that is smooth and elastic and passes the windowpane test. Quite often, the kneading will be interrupted to make minor adjustments to the flour and water used, and also by a period of rest, which allows the flour to more fully absorb the water.
The work surface approach usually calls for combining the dry ingredients on a work surface, making a well in the center, and gradually adding the water into the well while combining it with the dry ingredients, using either a fork or your fingers, being careful not to let the water escape from the well and make a mess of things. Once the ingredients have been combined into a rough ball, then the rest of the kneading is done by hand, until the final dough ball is smooth and elastic and passes the windowpane test. Again, the kneading process may be interrupted to make adjustments to the flour and water used and by a period of rest.
If you don't have a pizza stone, or you want to keep your costs down, then you can always use inexpensive unglazed tiles. There is a fair amount of material on both stones and tiles at this site at your disposal if that is the way you want to go. Just use the search feature. As alternatives to pizza stones or tiles, you can also use pans and screens. Again, there is a fair amount of information on this site as to the use of both pans and screens. Pizza stones, tiles, pans and screens all produce good pizzas, even great ones. Just start somewhere, get the feel for what you are doing, and, with time, you should do just fine. Remember, Confucius reminded us that a ten thousand mile journey begins with a single step