You can begin harvesting your sweet potatoes as soon as the leaves start to yellow, but keep in mind that the longer they are left in the ground, the more vitamins they'll have. After a frost the vines will turn dark. At this point, the tubers should be harvested as soon as possible to prevent rot.
On a sunny day when the soil is dry, use a spading fork to dig sweet potatoes. Take care not to nick the tubers because this will encourage spoilage. The tubers can grow out as much a foot from the plants.
Dry your harvest in the sun for a few hours, then bring them indoors and continue drying in a warm, humid place for a couple of weeks. An open area of in the kitchen is a perfect spot for drying them As the tubers cure, the starches are converted to sugars. Properly cured sweet potatoes will last for several months. For long storage, wrap individual tubers in newspaper and store them in a 55F to 60F basement.
For lots of information about cooking sweet potatoes see:
Glorious Gold and Orange by Leigh Abernathy