Hi Robin,
Welcome to the forums. Hopefully we can help you work through your issues.
The first thing we need to know is the exact G5 model you have. The reason for this is that in order to run Snow Leopard, OS 10.6.8, your machine must a minimum of an Intel processor (IA-32). From this link:
https://en.wikipedia.org/wiki/Mac_OS_X_Snow_Leopard#System_requirements
it looks like your machine should be OK.
Secondly, are you making backups, to an external device? If you are, wonderful, and if so, what software are you using for that? If not, then you will need to do that first before upgrading to OS 10.6.8. Given that you are running OS 10.5.8, you can download and use SuperDuper! V2.7.1 to make a backup/clone of your system to an external device. Once you get to OS 10.6.8, however, you'll need to upgrade to V2.7.5 of SuperDuper!.
Next, I assume you have a DVD drive with your G5. You'll need to purchase the Snow Leopard (OS 10.6) DVD from the Apple Store. It costs $19.99 US, and here is the link that discusses that:
http://www.apple.com/shop/product/MC573Z/A/mac-os-x-106-snow-leopard
Once you do that (after having done the SuperDuper! backup), you would boot your machine from the DVD. Next, you would need to use Disk Utility on that DVD to 1) Erase, 2) Format, and 3) Partition ( I don't know how many partitions you have now, nor how many you need) the internal hard drive. Then, you would install OS 10.6 onto your G5. Once that completes, boot your G5, and then use Software Update to upgrade to OS 10.6.8. Here's some info about that:
http://www.apple.com/shop/question/...-mac-os-x-1068-snow-leopard/QJF4KHAJ2P74YHAJT
Finally, after upgrading, you can use Migration Assistant (should be inside the Utilities Folder within your Applications Folder) to "migrate" (copy/move) all the non-system Files, folders, etc. from the SuperDuper! backup to the G5.
The only "concern" might be the "health" of your internal hard drive. When you run Disk Utility from the Snow Leopard DVD, it can first tell that information before you erase, format, etc. the drive.