The latest and best option for an operating system for your Power Mac G5 is Mac OS X 10.5.8 aka Leopard. If your machine did not already come preinstalled with Leopard, you'll need to procure a disc via eBay or another Mac reseller. Other options include Mac OS X 10.4 (aka Tiger) or a variant of Linux (like Debian). There is one instance where you might need Tiger for your G5 - if you need to run classic Mac apps alongside your other apps. Otherwise, Leopard is fast, beautiful, and good enough.
Apple still has a Leopard support pages here.
Last Updated: 4/23/2016
Onyx for Leopard is a free system utility for your G5, giving you lots of options to maintain, modify, and fine tune your machine. Many features simply apply a GUI to command line utilities and tweaks. You can do things like show hidden files in Finder, turn off or on various options, run scripts to improve performance, check on the health of your drives, and so on. A useful, nifty app to have available. Best of all, it's free.
Dropbox is an awesome backup/shared folder service that syncs with other Macs, Windows PC, and Linux boxes. This is a great way to share work and keep a simple backup of key files. Earlier this year, Dropbox announced that it would be logging out all computers earlier than Snow Leopard.
Good News: There is a workaround! Read this post and the rest of that thread (for an even easier solution). Dropbox lives!
It should seem weird that I failed to mention one of Leopard's best features - Time Machine. In truth, I rarely did serious backups except for an occasional copy to a USB drive until Time Machine debuted. It made backups on the Mac flawless and simple. The graphic interface to recover old files is also a fun little bonus (even if it is probably unnecessary). Make use of this feature. It will save time and sanity.
If you want to live a little more cutting edge, a Linux distro does offer a more updated operating system for G5s. For example, you'll have access to more secure browser options. There are some caveats to this, mainly the lack of favorite or equivalent Mac OS X applications and the inevitable rough edges that come from running an open source OS. There's a lot written about preferences between Linux and Mac OS X, the pros and cons, the ins and outs. I'd recommend you do some research on your own. I can only tell you what works for me:
I've got nothing against Linux - I use it as a server OS with a couple of VPS accounts. My home backup server runs a variant of Linux. It's useful, rock solid, and customizable. As a desktop OS, though, I've never been able to use it for an extended period of time without running into lots of annoyances, some minor, some major. Ubuntu is by far my favorite of all Linux variants, followed by the Mac-like Elementary OS. Even with the extra polish those distributions work to achieve, there is little that I can do in them that I cannot do faster and better in Leopard or El Capitan. The selection of applications can be a bit anemic, with user interfaces that might be serviceable or disaster-like. And one more thing - major system upgrades can require a complete reformatting and reinstall.
Again, do some digging, check out install instructions, and explore your options. You may find it is an adventure worth beginning.
Disco is a simple but great cd/dvd/ burning app. You can create ISOs, data discs, and other options. Simple, fast, elegant, and now free. The app features a cute (although unnecessary) graphical effect of spewing smoke as it burns your disc. The website lists a free registration code for anyone to use.
Burn is a another great option for burning cds/dvds, especially since it is open source and seems to remain somewhat updated. You may find it has more options for what you are trying to do. In truth, I can't remember the last time I burned a CD, but I'm sure there will be another opportunity in the future. How technology changes.
Carbon Copy Cloner is a fantastic little app that gives you lots of options for saving backups and cloning your disks. It's a classic tool for any serious Mac user. You can get version 3.4.7 from the link above.
Mountain Lion Theme is a fantastic visual enhancement for Leopard that makes your Mac look more like Mountain Lion. It's gorgeous. The link above includes both an installer and uninstaller, in case you run into problems.
Candybar is a slick theming tool for your Mac. Best of all, it is completely free. This came as a surprise to me, since I never felt like a theming tool was worth $30 back when I tried the demo. Newer versions of Mac OS have locked out Candybar's tricks, so the benefit is those of us who run older Macs. Grab it and enjoy.
The Yosemite Theme Pack can be paired with the Mountain Lion theme to bring a little more current flavor to your Leopard install. I have not tried this out yet, but it looks like it will be updated again very soon. I will be giving it a shot in the future.
iOS X is a cool theme that turns Leopard into a bit of an iOS 7 style look. It's not super exhaustive but includes a new font, some desktops, a new dock style, and so on. It actually pairs decently well with the Mountain Lion theme that I listed earlier. You will need Candybar to completely install and use the new fonts throughout your system.
iOS 7 Screensaver is a copy of the new iOS 7 lock screen as a screensaver for your Leopard Mac. It's pure eye candy, but it looks pretty. The bonus wallpaper included is also worth the entrance fee (which is free).