Sweet potatoes originate from tropical central and South America, but they grow very well year round in Perth (unless you happen to live in frost prone areas. In colder regions, they can still be grown throughout the warmer months.)
Choose a healthy looking whole sweet potato (organically grown if possible) and put the narrow end down into a glass of water. You can use toothpicks poked into the potato to suspend it. I used a hyacinth vase which worked very well indeed. Conventionally grown sweet potatoes may be treated with a growth inhibitor to extend their shelf-life - this will also delay sprouting.
Put the sweet potato in a well lit, warm spot (a window ledge is ideal). Mine took weeks to sprout, as I initially attempted to sprout it at the height of winter. It did nothing until the weather began to warm up, then it literally grew overnight.
The end in the water will start to put out roots, and from the section above the water, little shoots will form. These shoots (called slips) can be neatly snapped off (pull downwards) once they reach about 10cms in length and begin to form leaves. Removing slips will encourage new ones to form - you will get at least 20 from one tuber.
The slips can be planted directly into pots, or you can put them in a small glass or vase with the snapped off edge sitting in water, and within days each slip will start to put out roots of its own. I like to do this to ensure that healthy roots have sprouted, but many people go straight from the slip to soil without problems. Once the plants are about 30cms long, plant them into your garden.
Each of these will form a new plant, and in time (about 3 - 6 months) form a tuber to harvest. You can carefully dig around your plants to find a tuber. As vines grow, they will continue to set additional tubers, so the oldest tuber will usually be the largest. Carefully work your way backwards from the growing tip to find it! Look for where the stem is thicker and dig around.
When harvesting, try not to damage the tubers. Shake off the excess dirt but do not wash them. Leave them to cure, not touching each other for a week or so before storing away. The skins will naturally thicken up and heal over small scratches in order to protect the tuber.
As a bonus, sweet potato leaves and shoots can be harvested and lightly steamed as a tasty and healthy green vegetable. They are reputed to have anti-cancer properties.
Should you have a sweet potato left in the cupboard that begins to sprout, this too can be planted out; but you are still better off snipping off the slips and using them as cuttings in order to encourage the formation of new tubers on each one, rather than allowing the whole original potato to sprout and grow. As there is an existing tuber, this will delay formation of more.
Sweet potato vines can be quite sprawling, but make an easy-care ground cover if you have a large area. They require reasonably fertile, well-drained soil (poor drainage will make them rot). Avoid using too much nitrogen rich fertiliser on growing vines, as this will encourage more leaf growth and delay formation of the tuber.
Once you have sweet potato vines growing, you can easily propagate from the growing tips of your existing plants.
Sweet potato is a hardy plant that is relatively pest and disease free. Excess growth can be composted and used as a mulch, and of course the whole vine makes an excellent living mulch to protect and cool your soil. If you have the space you could consider a permanent patch around fruit trees, or even on your council verge! (Check whether this is allowed first.)
In colder regions, tops will yellow off in autumn, which is a good indication of when you can harvest the tubers. Replant in spring.