Cody is correct in almost everything IMO. Wood can be cut at any time of year. For woods like osage, I like to cut it(when I can find any)during the dormant periods(winter) but any time you have a chance to is OK. For white woods, I cut during the growing season so the bark will peel off without much effort. On whitewoods, what is under the bark will be a good back for your bow.
I consider hickory, red and white oak, maple, hop hornbean, cherry, walnut, etc, to be whitewood. I make bows with their sapwood.
With any wood freshly cut for self bows, split the log in half and seal the ends and store properly in a cool dry place. I usually cut staves(logs) to 6' or 7'long. If you peel the bark off of the white woods, seal the back. With osage, mulberry, locust, etc, Leave the bark and sapwood on, at least for a while, until some of the moisture is released. When you remove the bark, also remove the sapwood(unless you want a sapwood back for the color contrast) and seal the back. These woods WILL check badly as they season if not sealed.
I have never had wood cut at a sawmill but Cody's recomendations about that sounds good to me. Pat