Satsuki - tend to have a flush of growth in early spring, then they bloom, May-June, then followed by a second flush of growth. Buds for the following spring are usually set before the autumn equinox in my area, though they could set later for some varieties.
Kurume azaleas tend to go straight to blooming after winter dormancy, no vegetative flush first. This puts them one to two months ahead of Satsuki for blooming. After blooming, the growth happens. There are a fair number of hybrids between the two groups, so it is best to just observe the cultivar you have and keep notes on what it naturally does.
I usually thin the flower buds for my azalea, but I never can bring myself to remove all flower buds. The point of raising azalea is the flowers, otherwise one could just grow boxwoods. I hate boxwood, I find them boring.
In autumn, I decide if I will be repotting & doing drastic work on an azalea. If I am going to do drastic work, I remove all, or just about all flower buds. The azalea will need its strength if I am going to do severe work on it. Years where I know I am only going to do wiring and pruning, I let them flower fully, or with just a light thinning of flower buds.