Backup just your users files off the machine before doing anything and unmount and disconnect the drive.
Upgrade to Snow Leopard and run the Software update until clear
Provided your machine is working good, Snow Leopard is a darn good upgrade for a 10.5 box. Especially if it's freshly installed.
Do not upgrade that older machine to OS X 10.7 or later (or to Mavericks 10.9) it's too out of date (old), it will be slow and your PPC based software will no longer work. (will under Snow Leopard/Rosetta)
Currently 10.6.8 is fine with your iOS devices and geting security updates, we still have 25% OS X market share still, but eventually a new machine is likely in order for you soon.
I advise breaking in a new Mavericks 10.9 machine, avoid upgrading that older one past 10.7, use it for backup or sell it.
OS X 10.4/10.5 need to upgrade, 10.6.8 ok still