Garlic is one of the simplest things to grow, DC. Here is the general proceedure. Then we can talk about your specifics.
First, most people fall plant garlic. Depending on your specific conditions, it might not be too late. I'm not sure if it's late fall by you, or already winter? If you can still work the ground, you're ok. All the cloves need is enough time to put down a root structure. Then they'll go dormant over-winter. I've planted them here as late as January, and they've still grown. Sometimes conditions let them actuall sprout before winter sets in, and the leaves get frost bitten on their tips. This is not a problem, and doesn't affect later growth.
Garlic is a heavy feeder, so prep the ground well. I put down a good base of compost, then amend further with bone meal, dried blood, and wood ashes at the rate of 1 cup each per ten row feet. This is turned in well. Long about May (roughly halfway into the growing season) I sidedress with the same amendments.
Keep the bed well weeded. Garlic doesn't like competition. And keep it watered until about a month before harvest, at which time, stop watering.
Depending on variety, the maturation times for garlic is 120-150 days. Bulbling, itself, takes place during the last month of that growth.
Sometimes, due to growing conditions, the plant produces a single, relatively small, undifferentiated bulb. This is called a "rounder." It is perfectly edible. Or you can cure it like a regular head and replant it. The following year you'll get a large, differentiated head.
If you continue using store-bought garlic I would not wait until it sprouted. Buy some specificly to plant. Once you get garlic going it's a self-sustaining crop. But, frankly, I would look into buying some of the so-called gourmet garlics and try growing them.
BTW, while the differentiated bulb is the main goal, don't forget the other edible parts of garlic. The young plants---green garlic---can be used just like scallions. And the hardneck varieties put out scapes, which are a culinary treat.
