How to load using AJAX, passing data in advance via POST?

Question

AJAX

I'm learning to program, I immediately took up a live example. Help to implement, please.
The bottom line is that using AJAX (in order not to reload the page) I pass the value from the form to the php file using POST, get the result and insert it under the form.
I don't make crutches, is this the right decision?
Right now I just made a php script that gets the value and outputs the value. Please suggest suitable manuals.