Potatoes are fun and easy to grow. The potatoes grow in the soil, while the green leaves are the only thing that show above the ground. At the end of the season, when the plants have died back it's time to dig up your bumper crop of delicious potatoes.
In the NY/NJ area potatoes are generally planted in April or May, though you can plant them through the first two weeks of June for a later crop of potatoes. In locations further south you will be able to plant them earlier. In Southern California and South Florida, they are actually often a winter crop!
The important thing is that the soil temperature be at least 45 degrees Fahrenheit about 4" below the surface of the soil. Below that your potatoes won't grow as well, and the seed potatoes may rot before sprouting. You can take your soil temperature with an Instant Read kitchen thermometer.
NOTE: If your seed potatoes are large, you can cut them into pieces before planting. You need at least one eye per piece. They should come with instructions for dividing. Do not use grocery store potatoes even if they have sprouted. They have often been treated to suppress growth and may carry plant diseases you don't want to spread into your garden.
I have two methods I recommend for growing potatoes. One is in a potato grow bag; the other is in a raised bed.
This is the potato grow bag I recommend from Gardeners.com. There is a nice video on the product page showing how to grow and harvest potatoes in a grow bag. The grow bag method is a fun and easy way to grow and harvest potatoes. The yield is slightly less than growing in a raised bed and you may need to water more often but the ease of use makes up for that for many gardeners. You can reuse your grow bag in subsequent seasons but should not reuse your potting mix. You can pour it into any raised bed where you will not be growing anything in the nightshade family (potatoes, tomatoes, peppers, eggplants) the next year, and buy new potting soil.
In a grow bag you plant 7-10 seed potatoes evenly spaced in your bag about 4" deep with the eye or sprout facing up. Start with your grow bag about half full of potting mix. As the potato plants grow to about 6"-8" tall add potting mix, leaving the tops of the plants sticking out of the soil. Do this a few times in the season until your grow bag is full. Potatoes are harvested when the foliage dies back. To harvest potatoes from a grow bag simply pour the contents of the bag out and search for your potatoes.
To grow potatoes in a raised bed, you want beds at least 1 foot deep. Your potato bed should not be filled all the way to the top of your bed. Leave about 4"-5" of space to add soil later. Plant your seed potatoes about 9"-12" apart in all directions, 4"-6" deep, with the eye or sprout facing up. Once the plants grow to 6"-8" tall you will add soil to your bed to bury part of the stem and leaves and encourage more potato growth. You can do this two or three times throughout the growing season. Possibly the second, and for sure the third time you will likely be mounding up soil higher than the edges of your raised bed. That is fine. You want the extra soil to be added around the stems of the potato plants. Use a "raised bed mix" or potting mix to do this, not heavy topsoil.
Potatoes in a raised bed are also harvested when the leaves turn yellow and die back. It's a bit more of a treasure hunt to harvest potatoes from a raised bed than from a grow bag but the yield per square foot is higher. I like to use both methods. I'd suggest you try both if you can and see which you prefer!
Here are some resources for ordering seed potatoes:
You can also look at your local garden center