If you want to grow garlic, buy some "seed garlic" instead of using leftover cloves from the grocery store. It will still just be regular garlic, and you'll plant the cloves, but sometimes grocery store garlic is treated to prevent sprouting. You'll also notice when looking at a catalog that there are many varieties of garlic, and you'll be able to choose a variety that is most suited to your local growing conditions.
Select a planting site with deep, loose soil. pH will ideally be about 6.5 -- similar to what other garden vegetables like.
Plant garlic in autumn before the ground freezes -- but not too early, you don't want it to put out any top growth before winter. Mulch with a thick layer of straw or autumn leaves. (Last year I used about 8" of leaves with good results.) Plant only the larger cloves, which will produce larger bulbs. Save the small ones for eating.
In the spring, pull back the mulch before the weather warms and the plants try to send up shoots.
See this question for information on harvesting.