Considering that some of the winter squash can last right through winter and into spring, there really is a squash for every season. By the time the last of the winter squash run out, the first of the summer squash will be ready later in spring. They get an early start here, and continue until frost. By that time, the earliest of the winter squash will be ready for autumn, and will last through winter.
However, they all get planted about now. There is no such thing as cool season squash. They are all warm season vegetables (actually fruits). The vines and foliage all grow like weeds while the weather is warm, and then succumb to frost at the end of their season. They all grow very easily from seed, but because each garden needs only a few plants, they are often planted as seedlings.
The main difference between summer and winter squash is that summer squash plants continue to produce many smaller fruits while the weather is warm, and winter squash plants produce only a few larger and firmer fruits that develop slowly while the weather is warm, and then finish ripening as the vines that produce them die in autumn and winter. One begins where the other finishes.
Zucchini is the most familiar summer squash here. It is most productive if regularly deprived of its tender juvenile fruit, and is notably less productive if the fruit is allowed to get bigger and tougher. Pumpkin is the most familiar winter squash here. It is expected to produce only a few big and firm fruit that take all season to develop. Extra juvenile fruit may get plucked to favor one or two fruits.
Summer squash deserve more prominent locations in the garden because the fruit gets harvested regularly. Zucchini grows as a large plant, but stays somewhat confined. Winter squash want the same sort of rich garden soil and watering, but can be put out of the way. Except for plucking a few extra fruit, not much more needs to be done to them. Some like to climb fences. Pumpkin vines can be directed to the edge or just outside of the garden.