Prep the Potatoes and the Soil
Start with certified quality seed potatoes from your local nursery or gardening center. These are not true "seeds" but are modified stems called "tubers." You can try using potatoes from the grocery store, or from last year's crop, but the risk of getting an inferior crop is much higher.
Potatoes may be planted in early spring, as soon as you can start working the soil. The soil should be worked well, with organic material deeper than the actual plants. The organic material may induce disease if placed directly on the potatoes, but will help the root system to grow. Do not use manure, as it may cause scab. If your soil is heavy, try planting in raised beds.
Do not waterlog your potatoes or the soil around your potatoes at any time — this causes potato rot and will compact your soil.
One or two weeks before you intend to plant, set your seed potatoes out in a warm, dry place where it will reach 60 to 70 degrees. This will help them to begin the sprouting process. One or two days before planting, cut the potatoes into squares about 1 1/2 to 2 inches across, with one to three "eyes" to each square. If your seed potato is small, just use the whole thing. Make sure the potatoes have formed a hard callous where you cut them — this will help prevent rot when they're planted in the ground.
Plant the Potatoes
There are two ways to plant potatoes. The first method is in mounds, which works well if you have a smaller planting area. Dig a hole 6 to 8 inches deep, with a diameter of 3 to 4 feet. Plant 6 to 8 potato seeds in this area cut side down. Cover with 4 inches of soil (at this point, you'll still have a hole in your garden).
When the sprouts grow about 6 inches tall, cover them with another 3 to 4 inches of soil. Your potatoes will grow in this space between the seed and the surface. When the sprouts grow another 6 inches, it's time to add another "hill" to the mound, adding about 4 more inches of dirt. A good rule to follow is to "hill" whenever your sprouts reach 4 to 6 inches, stopping when they begin to flower.
Over the next few weeks, it may be necessary to add 1 or 2 more inches so the potatoes don't peek out of the ground. When exposed to the sunlight, potatoes can turn green, which is toxic. Don't cover up too many of the sprouts — leave enough on the top so that it gets energy to grow.
The second method is to grow rows of potatoes. Follow the same instructions as the mound method, but place your seed potatoes apart every 15 inches. The row method demands quite a bit of space.
Water your potatoes well during the growing season, but especially when flowers appear on the plants. This is when the actual potatoes are forming. Water the plants early in the day so they have a chance to dry during the day.
Harvest the Potatoes
Harvest your potatoes two to three weeks after they've finished flowering (the stems get yellowed and brown). Stop watering at this point to give the potatoes time to dry. Carefully dig up the tubers, especially if you're using a shovel. Give them a chance to dry for two to three days in the dirt or in the garage/gardening shed. This will allow the potatoes' skin to thicken so they store better.
Due to pest control, it's best to move your potato crop every year. You can re-plant an area every three years.