Look for firm potatoes with no brown or green spots, sprouts or eyes.
Brown paper wrapping is preferable since it keeps out light.
Choose the variety depending on use:

"mealy" for mashing, French fries and soups
"waxy" for salads, gratins, scalloped potatoes
new potatoes to serve on their own

Steam, boil, braise, sauté, etc. - potatoes should be cooked til tender in the middle, but not be overcooked.
Add 1 Tbsp. white vinegar to the cooking water to keep potatoes white.
Never leave a raw peeled potato sitting out, since it will oxidize and blacken. Put it into salted water with a little lemon if you are not ready to cook it.

Cooking potatoes in their skins
There is, of course, the traditional baked potato, but you can also remove the cooked potato flesh and whip it with milk, butter and mascarpone or goat cheese.

To make a wonderful potato galette, boil the potatoes in their skins til three-quarters cooked. Allow to cool, peel, grate and brown in a frying pan in a mixture of butter and oil.

Gourmetpedia
Grated and cooked as little cakes, potatoes become "rösti," a traditional accompaniment for sausage or veal escalopes. They may also be served with onions and/or bits of bacon.