Depending on your zone, you might want to plant garlic now to harvest in July or August next year. You can go to the grocery store and get a garlic clove and it will usually grow just fine, despite what people say about them being treated with growth inhibitors. If they sprout in your pantry, they'll grow in the ground. But if you're worried about chemicals on your garlic (even though you take all that off when you peel it), then buy organic cloves. Look for nice fat ones with no soft spots. Most garlic sold in grocery stores is hardneck, and does not store well raw, so use it soon after you buy it, or preserve it in some way. Hardneck garlic has flowers which are borne on a stem that comes from the center of the bulb. The flower stem is rigid and comes up through the 'neck' of the bulb, thus the name.

Softneck garlic is the 'storage' garlic. It does not have flowers (very rarely, anyway, depending on the cultivar, and requiring an absolute harmonic convergence of many factors), and so does not have the rigid flower stalk, thus the name. This is the one you see braided, which you can do because it has no flower stalk. Softneck garlic is usually available in garden nurseries as starts.

Of the two types of garlic, softneck is more mellow-flavored and hardneck is usually more pungent. Look on the internet for a garlic festival in your area. There is often a nursery that grows garlic in attendance and you can order cloves from them. In my area, the local garlic grower has almost a hundred different cultivars, and some are pretty pricey. Connoisseurs can tell the difference between them, but for someone who's never grown it before, go cheap and shop at local sources.

Elephant garlic, by the way, is not a 'true' garlic, but is a closer relative of the leek. It's fun to grow, too, and you can sometimes get cloves of it at the same places where you buy your bagged spring flower bulbs. Recently, more growers have also begun bagging and selling both the hardneck and softneck garlic this way, too.