Garlic is very heat resistant, so it's easy to grow even in Arizona. Garlic only needs cold temperatures in the first few months for optimal yield. Plant it in late October and it should do just fine outside year round. You can go through the trouble of starting the growth in the fridge, I don't think it's worth the extra effort. But if you really want to start now put the cloves in some water for 1-2 weeks until they sprout and have roots. Then put the garlic in smaller containers and keep them in the fridge for at least 2-3 weeks and up to 2 months. Then transplant them outside in bigger containers. While in the fridge you need to water and moisturize the garlic a lot as a fridge is like a dehumidifier.

You want soft-neck garlic in hot climates. Which is the most common type found in grocery stores so the ones you currently have should be fine.

For the soil, you want something that drains well. Loam soil is a good choice. You also want to mix it with a decent amount of compost, somewhere around 25-40%.

Space the cloves out a fair bit. We do about 15 cm/6 inches. Watering really depends on the circumstances. At first, they need rather little water, so maybe water them every 7-10 days if it doesn't rain. Once they are bulbing you want to water them every 3-5 days. But just check the soil, if it gets too dry water it.