Thanks very much, Chris.
I truly hate to be the bearer of bad news but the last 3 major releases of OS X do NOT support Power PC processors.
In light of this fact, the newest version of OS X that can still be installed on your G5
is OS X 10.5 Lepoard.
Going from OS X 10.4.x to OS X 10.5 was a paid upgrade but it is no longer officially sold in stores and if you can't readily find it some customers have been able, in the past, to get it via telephone.
If you wish to perform this upgrade, you can call 1-800-MY-APPLE (1-800-692-7753) and tell them you want the 10.5 Leopard upgrade which they hopefully can send on CD for around $20 USD shipped.
Once you have the disc, the upgrade process is fairly simple. After the install, keep running System Preferences > Software Update until no more updates are offered.
If you have a software requirement for OS X 10.7 or 10.8 then, I'm afraid to say, the only option would be to upgrade to a newer Mac.
I'm sorry I don't have an easier answer for you but we are at Apple's mercy. If you continue to have questions on this topic, just reply back to this discussion prior to rating. I will be more than happy to work with you further. Hopefully, this info will answer your question and if so please don't forget to positively rate my service
Thanks and good luck,