Flatfile is only insecure if you have the files within the web document root otherwise its just slow.
I've heard tell that PHP has pretty powerful database and query functions - although strictly speaking it is nothing like MySQL
Umm.. if your saying PHP is a database your greatly mistaken PHP is a web programming language which has database functions to interact with many databases. MySQL being the most popular and it also getting its on category of functions to interact with the database.
As for starting out with an online booking system. Will this involve the exchange of money if so, I would be very careful and make sure you have a fair bit of knowledge within PHP first and Ecommerce.