Before planting sweet potatoes in your garden, you'll need to get your hands on some seedlings, known as slips. You can grow these yourself starting in January from an organic sweet potato, or you can buy them from a garden center, seed company, etc...
To grow your own slips, start with an organic sweet potato. Place it in a warm spot in your kitchen that hopefully gets a couple of hours of sunlight, at least. When it begins to sprout, stick the end in a jar of water, using toothpicks to keep it from touching the bottom. The bottom of the potato will sprout roots, and the slips will emerge from the top of the potato after a couple of weeks. Keep enough water in the jars as the potatoes get thirsty and drink quite a bit.
Once the slips are 6-8 inches long, you can pull them off the potato
gently with your hands. At this point, you can plant the slips directly
into your garden if it's ready and if the weather is cooperative. If
you need to hold off for a couple of weeks, put the slips in a jar and
add a few inches of water. In a few days, the slips will start to send
out roots. Keep the roots of the slips in water until you are are ready
to transplant them into your garden.
When planting sweet potatoes in your garden, wait until at least 2 weeks after the last expected frost in your area and until the soil has warmed up to 70 degrees F or so. If you have a short growing season, you can cover the ground where you want to plant with black plastic. Do this a couple of weeks before you plant, to heat up the soil.
Sweet potatoes grow best in ridges. Till the soil as deep as you can. Mound up dirt in rows, making the ridges at least a foot high and 2-3 feet across. If planting multiple rows, space the centers of the ridges at least 4 feet apart. It should be noted that sweet potato vines will begin to spread and by mid-summer will engulf any other variety of plant that is within 6-8 feet of the sweet potato sprout, so plan accordingly. Transplant the slips 12-18 inches apart in the row and cover the root ball with soil. The tops of the slips should be visible above the soil. Water the transplants in well.
Now that you're done planting sweet potatoes, it's time to think about watering and fertilizing them.