When you get past all the hype, all AJAX is is a way to use JavaScript to send HTTP requests and process the responses without having the browser have to load a new page. This is not necessarily "better" or "worse" than the more traditional way of of sending new page requests for each transaction. (Hmmm...can you actually use the word "traditional" when discussing anything to do with the Web?)
AJAX can provide a more seamless user experience, with the drawback that it depends on the browser running JavaScript, so you need to either provide a fallback mechanism for users who are browsing without JavaScript, or simply decide you don't care if that small minority cannot access the functionality provided by your AJAX implementations.
Security-wise, there is no reason I can think of that an AJAX application should be any more or less secure than a non-AJAX implementation.
If you are new with PHP, I'd probably recommend staying with "traditional" PHP page requests unless/until you run into some desired functionality that cries out for an AJAX solution, then start learning how to do it. (It's really not all that difficult: basically using a JavaScript function to do the HTTP request/response handling, and there are hundreds of them out there that you can pretty much copy-and-paste into your pages as needed.)
