If I'm planning on self-publishing an anthology of unpublished short stories that I wrote, do I also need to copyright each story, or will the entire anthology's copyright suffice for each story?
I am not a lawyer; but copyright law is clear: Everything is copyrighted, including individual stories within your work.
Read about Fair Use, but everybody is subjected to that. The same link, from the Stanford University law school, has a great deal of stuff on copyright law written for laymen to understand.
If you live in one of the 175 countries that are signatories to the Berne Convention, then your work is automatically copyrighted when you create it. Registration is not required to have copyright to your work.
There are still advantages to registration in some countries. In the US, for example, you can't sue for copyright infringement in federal court unless you have registered your copyright. For unpublished works, registration can establish the date by which you had created the work, which is helpful if you claim someone infringed your right and the other party says he got there first.
Because registration simply establishes your claim, there's no need to separately register stories in a compilation that will all be published for the first time together. If you want to publish some of them at different times, and you want the protections that come with registration from the earlier dates, then you would need to register those works separately.