How much of someone else’s work can I use without getting permission?

There is no fixed amount. It depends on the usage..

“Under the fair use doctrine of the U.S. copyright statute, it is permissible to use limited portions of a work including quotes, for purposes such as commentary, criticism, news reporting, and scholarly reports. There are no legal rules permitting the use of a specific number of words, a certain number of musical notes, or percentage of a work. Whether a particular use qualifies as fair use depends on all the circumstances. See FL 102, fair use, and Circular 21, Reproductions of Copyrighted Works by Educators and Librarians.”
— U.S. Copyright Office