You know you need permission to use that quote/ graph/ picture/ song lyric, but how do you ask? Get started with the list I give today in my How To column at Copyediting.com.
There are editors who specialize in securing permissions, often they are production editors, involved from the get-go of the project. Many academic publishers require authors to secure their own permissions, and pay for any rights, too.
(When) You Need Permission has two interactive guides to determining fair use/ dealing. There are significant exceptions; you may not (legally) have to ask permission at all. But it is polite and maybe even good marketing. Also see Copyright and Permission and the Project Checklist & Timeline.
You’ll find lots of helpful guides online. Try searching “copyright permissions” or “sample permissions request” and you’ll find resources like these:
Here are tips for getting permission to use lyrics in your book.