Actually the mysql documentation looks pretty good to me.
It's got a table of contents you can click on.
After that, it has a function index AND a concept index.
In addition, it has a search function.
I can always find what I need on that site in no time.
As far as searching for the API for PHP, that involves PHP, and not mysql so much.
Any PHP book should tell you how to do it, in fact, on this very site you can search the PHP manual for mysql functions.
How PHP connects to mysql involves PHP, it's not mysql's responsibility, and it's generous of them to even include language specific functions.