Again, it depends on your circumstances. You'll find that is one of my standard replies.
For some types of coops, I think shavings are best. For a walk-in coop where you have to occasionally clean them out, straw can matt together and be real hard to remove. For those little elevated coops where you can move a wheelbarrow under them and rake the bedding into them, straw might be best or maybe shavings will work just as good.
For nests, I have used straw, shavings, and hay. They all work, but again, for some of us, I think one may be better than others. We really have a wide variety of ways we handle nests. For any of them, you have to have a high enough lip to keep them from scratching the bedding out. I kinda prefer the straw or hay since I think they hold the nest shape better than the shavings, but I have had perfectly good results with shavings. Some people use carpet, Spanish moss, or shredded paper. I'm surprised I have not seen someone mention regular dried moss.
I don't use anything for my run, just bare dirt. If you have drainage problems, it is often a good idea to put sand or a gravel bed covered with sand to improve drainage. If you have a big enough run or normally free range them, you may never have to rake or clean a run. Some people rake their runs daily. I don't suggest using organic material in the run, such as shavings, straw, or hay, but some people do. I'd worry about the stuff getting wet and molding here. Mold is very bad for chickens. But some people have developed a strategy for handling that.
I don't do house chickens.
You'll find we do have a wide variety of ways we handle things, based on personal preference, what is locally available, personal experience, often only limited by our imaginations.