Potatoes are grown from potatoes harvested the previous year. They are called "seed potatoes". You may have noticed that sometimes potatoes left in your pantry start to sprout? This is them getting ready to reproduce.
It is recommended that you purchase seed potatoes from a nursery or from other growers, and do not use supermarket potatoes.
You can get your seed potatoes ready for spring by any of the above methods. However, I never bother and simply plant them straight in the ground before they have sprouted. If you do have a potato which has sprouted, you can cut them up leaving one shoot on each piece to get more potatoes for planting.
Plant in spring after the last frost. However, I often plant mine out earlier. It just means that they either won't sprout until it's warm enough, or any sprouts may get frost bitten. This doesn't effect the tubers below the soil. The main take home is they are a summer crop and frost sensitive.
Potatoes grow by the tuber (last year's potato) sprouting a little tree. This potato seed is as deep as your potato plant grows. The stem grows up and above the soil. Between the seed potato and above the soil, new little potatoes sprout off and grow. In order to encourage more potatoes to grow, many people "mound up" their potatoes. This means as the steam grows, they raise the soil leave up the stem. But always leave 15cm of the stem above the soil. The stem that has now been buried will sprout more potatoes. The more you mound up, the more of the stem that is below the soil, and the more new potatoes that sprout off the stem. This growing method is often why people grow potatoes vertically in structures like towers. But you can grow them in anything, even your garden beds.
Choose your planter
This also applies if you have decided to get the jump in spring and planted your potatoes indoors.