I trying to find the best way to upgrade from PHP 5.1 to 5.3. The CRM software I am running on this server requires this upgrade or else I probably wouldn't even perform it, because it seems like it's going to be perhaps trickier than I hoped it would be. Being still new to the programming world, these routine upgrades are still worrisome to me.
I am running apache 2.2.6 (Fedora), PHP 5.1.6 and MySQL 5.0.27 on this server.
From the php version i gather that you're using Fedora core 6, which is way out of date. You should investigate upgrading to a OS where you can still get security updates like RHEL or Debian stable. Fedora is not really suited for servers due to the short support time.
A quick alternative to the upgrade is to install Zend Server, which provides with a standalone xAMP stack, including either PHP 5.2 or 5.3.