There's not a lot of information you need to grow good garlic, but these few tips will help you grow it better.
1. You can use cloves of bulbs from the grocery store.
I have heard that some producers treat their garlic so it will not sprout, though I have never seen this. If you're unsure, look for garlic that is already sprouting or buy from your local provider.
2. And, you can replant what you grow.
In this sense, you can buy garlic once and have it forever. I have heard that some of the fancier varieties, like Russian Red and Elephant, may lose some of their flavor over many seasons, at which point you may want to buy new. I just grow plain old garlic.
3. Your garlic will either grow a 'stiffneck' or 'softneck', the differences being:
Softneck- as the name implies, the tops of these types are soft and can be braided, which may seem like a nice feature in your kitchen, except you are supposed to store garlic at near freezing temperatures. This is the most common garlic grown for groceries.
Hardneck- produces a swirly 'scape' that should be removed so the bulb below will grow larger. The scapes have a wonderful garlic taste and are great added chopped to salads or in cooking. Hardneck garlic is better if your winters are very harsh, is easier to peel, and said to taste better. It produces less and does not hold as well as softneck.
4. Plant garlic by putting one clove in a hole about 2-3" deep and 6" apart, root side down in a good well drained soil. In Zone 5 we plant about mid October, generally around the time the first fall frost is expected. Don't stress it if you see some greens before the snow falls, garlic will be fine over the winter and regrow the greens come spring. Mulch will help control weeds and keep your garlic a little warmer over the winter.
5. Harvest garlic when the tops begin to turn yellow and fall over. Here that is often late July though it is supposed to be mid August. Follow this crop with some cool season veggies such as broccoli and cabbage.
Botanical name: Softneck Allium sativum
Hardneck Allium ophioscorodon
Yield: Each clove produces a bulb.
Days to maturity: 9-10 months
Harvest: When softneck tops fall over, or at 9 months for hardneck varieties.
Storage: Hang and keep in a cold spot, or refrigerate. Dehydrate to make garlic powder.