I'm looking into using Ubuntu 13.10 server, but was noticing the apt-get PHP version that installs is 5.5.3, when PHP 5.5.9 is the most current version according to php.net. I'm wondering if I can trust 5.5.3 to be secure or if there is a trusted way to keep PHP more current.
Only you can decide and to do so you have a bit of work to do. You'll have to track down all of the security related bugs in the change log and then check to see if they have been backported into the 5.5.3 provided on Ubuntu 13.10. You can find the Ubuntu Security Notices here.
For example USN-2055-1 is a security related bug that was fixed in php 5.5.7 but as you can see from the USN it has been backported into all of the currently supported Ubuntu versions.