It sounds like a decent way to start. The main goal would be to reduce the effect of the clay on your soil, and it sounds like you have the right course of action in mind.

I am going to plant my garlic this fall in two of my raised beds for the first time; I have heard that raised beds work well, and am convinced enough to try it for myself.

Raised beds are fine, great, in fact. For soil put in as much compost as you can get your hands on, they could be grown in straight compost quite well. Otherwise using clay based soil is fine too, but it has to be kept loose (compost again).

The way I planted garlic in clay ground when I started years ago was to rake up the hills again after digging potatoes. Really raked them high and then just before planting garlic in the fall I would till the hills down to about 5 inches high and about 16 inches wide. I would make a trench and throw in leaf mold and bonemeal. I would fret about the clay soil preventing the bulbs from expanding to their optimum sizes. I would fret when I would see the early growth in the fall get frozen stiff and turn to mush in the spring. They always started growing and always produced nice bulbs, just as big as I get now in raised beds with sandier type soil.
The fun in growing garlic is the wondering of what is under the soil. Once you experience a few years of seeing that all the fretting came to naught, growing garlic became boring and every year now just dig trenches, throw some goodies in the trench, mix well, cover trench, plant biggest cloves 6 inches apart, mulch 3 or 4 inches with shredded leaves and just watch them grow. When 30% of leaves turn brown about 9 months after planting, dig them.You do have to snap off the scapes about a month before they are ready to dig. It's not rocket science. It's a boring crop to grow once you see that all your worries never come to fruitation. Geezer