Potatoes come in many varieties. For gnocchi, mashed potatoes (which are very common in Italy), fries (another very common fixture of the Italian table, despite they're being called French Fries in English), and dishes in which the potatoes are riced and then used as an ingredient you'll need mealy potatoes, which are high in starch and are also known as bakers; if you are instead planning on boiling them, putting them in a stew, or using them in salads and such, you will want waxy potatoes with a low starch content, which are also known as boilers.
The simplest Italian use for potatoes is Patate alla Ghiotta, which draw their name from the ghiotta, the pan used to capture the drippings from a roast: Peel your potatoes, dice them, and slip them into the roasting pan when you're oven-roasting a chicken or pork (in particular ribs, loin, and sausages); salt them and turn them occasionally. They'll absorb the drippings from the meats (boilers are ideal here) and emerge extraordinarily tasty. When to add them? About 45 minutes before the meat is done.
A Few More Italian Potato Recipes:
Patate Fritte alla Romanesca, Roman Fried Potatoes
Ravioli di Patate e Salsiccia, Potato and Sausage Ravioli
Patate al Forno alla Lucana, Lucanian Baked Potatoe Casserole
Patate Pasticciate, Piemontese Potato Casserole
More About Potaotes, and More Italian Potato Recipes
And Elsewhere on About:
Mum's Clam Chowder
Bavarian Potato Dumplings
Health(ier) Thick-Cut Fries (Friet)