When it comes to potatoes, there really isn't much to growing them. All you need is the a potato, some soil, and a planter big enough to accommodate it.
Potatoes should be planted in mid-February to early March. I planted my potatoes about 3 weeks ago. Then there was an accident and one of the planted potatoes was removed from the planter. Caution: Do not let curious children play around planters.
Potatoes are pretty fast growers. According to my notes, the time from planting to harvesting is on the order of 90 days. They require very little care, and grow easily in soils that would not be suitable for other plants.
At the home center, they sell bags of "seed potatoes". These are specially selected spuds that are superior growers. I have no idea how they determine this. From the looks of it, they are mostly misshapen or small potatoes that wouldn't make the cut in the grocery store. While cheap, I don't really need a whole bag of seed potatoes. I just need one.
My wife told me not to use her potatoes for planting because it was a waste, so naturally I kept it a secret when I snuck one kita-akari potato for planting. To prepare the potato for planting, I just cut the potato in half. There are other things one can do to promote growth like applying rooting powder, but I don't have that on hand and I don't grow so many things from cuttings, so it would just be a waste to buy it specially for potatoes.
Potatoes need about 30cm between each plant. My planter isn't quite big enough to hold two potato plants at that distance, so I've crowded them with only 20cm of distance. I'll probably pay the price for this later.
To plant, just put some rocks in the bottom of the pot to aid drainage, and add about 40cm of soil. Place the potato pieces (halves in my case, but you could cut a large potato into quarters as well) open face down at the proper 30cm spacing and cover with 5-10cm of soil. Water thoroughly and leave it alone.
Here is a picture of the current state of the potato that was dug out of the soil. I've circled the visible roots.
Note that the potato is facing upwards. This is because it was replanted incorrectly. However despite this, the potato is hardy and able to continue growing.