You do not need any special equipment to make pasta, but having the right equipment can speed up the process. A pasta machine and/or dough mixer are helpful, but we'll discuss making pasta with or without these tools.
Step one is to make your dough. You will need a mixer with a dough hook and/or a surface such as a wood counter board or marble countertop to knead on. In terms of ingredients, pasta is simple, but you do have some options you may wish to consider. Many people prefer semolina flour, while some people just use white flour or a mixture of both. Also, some prefer using eggs, while some just use water. If you are going to dry the pasta, you will want to substitute water instead of eggs.
If you have a mixer with a dough hook, add the ingredients and mix away. Otherwise, in a large mixing bowl, on your countertop or on a wood board, make a well in the center of a pile of the flour. Add the eggs or water into the center of the well, and SLOWLY mix the liquids into the flour with a fork. Then, by hand work the dough into a ball and prepare to knead. Kneading the dough is important. You will want to knead for about 10 minutes until the dough is smooth, soft and elastic. When you have a smooth ball of dough, cover it with plastic or a damp cloth and let it rest for about a half-hour.
Now comes the fun part. Cut a small piece of the dough. If you have a pasta machine, run it though the rollers several times, adjusting the thickness until you have a thin sheet. If you are rolling by hand, flour the surface of your counter or board and roll it out with a wood rolling pin until you have a thin sheet of dough. You can then decide what kind of pasta noodle to make.
Cutting by hand gives your pasta a real homemade look. To hand-cut, you can fold your sheet into thirds, and use a chef knife to cut pasta of your desired width - about 1/4 inch cuts for fettucine or very wide for lasagna, for example. Pasta machines come with various attachments you can feed the sheet of dough through for speedy, uniform cuts of spaghetti or other noodle widths. You can hang your noodles to dry on a wooden dowel or pasta drying rack. After you have cut all your pasta, bring a large pot of water with a tablespoon of olive oil to boil and cook until tender.
Ravioli can be made with a dough cutter by cutting squares, adding a small amount of your filling (try ricotta, romano cheese, salt, egg) and folding the dough over the filling. Then, cut about a half-inch perimeter around the filling and seal the ravioli by pressing the edges with a fork.
Mix the dough as described above. Form 1 inch, flat strips and feed them through your cavatelli machine. To make cavatelli by hand, roll small pieces into long, thin tubes (about 1/4 inch). Cut it into pieces just under an inch long, and form the cavatelli into little 'canoes' by pressing and then pulling and rolling with your thumb or a flat pastry cutter. You'll want to let them dry a bit, then freeze them on a tray or refrigerate in a ziplock bag.
If you are going to make homemade pasta, you probably also want to make homemade sauce. Sauce is something you can make to your personal taste using various combinations of fresh herbs, vegetables, tomatoes and olive oil. Use a stockpot or sauce pan, pour olive oil to coat the bottom of the pan. If desired, blanch some chopped onions and add garlic. Add chopped tomatoes and cook them down into a sauce with fresh basil, parsley, oregano, and a little thyme and salt. You may also want to add a can of tomato sauce and sugar for sweetness.