If I didn't make this abundantly clear before, I'll go ahead and do so now:
If you need to browse the Internet, I recommend you set this computer aside and buy a modern one. Old systems like these are a big risk on the Internet, because they're vulnerable to being used as part of a service amplification attacks, and because problems often go unfixed.(among other things, it has been discussed at length on this forum).
It's less about the security of your individual information (which most attackers don't want, or they'll just use phishing to get), and more about creating an Internet that's less likely to see spam and have DDOS attacks.
There are some, but not many, and no good centralized resource for hardening old PPC versions of OS X. The US Government has some hardening information, but it was published when 10.5 first came out and doesn't have updated information about, say, installing the relevant patches for, say, ShellShock, which Cameron Kaiser published.
So, my advice?
Get an Intel-based Mac, or some other sort of Intel-based computer on which you can run the newest versions of Mac OS X, Windows, or an up-to-date Linux (or other) OS distribution.
In terms of Linux on PowerPC, I don't have any specific recommendations. I haven't looked at that for years, because to me, a Mac is better used as a Mac. If you want to use OS X for some OS X apps, disconnect it from a network that can be routed to the Internet, or run OS 9 on it.
A modern, patched linux will be much more secure, but it will reveal the same thing tenfourfox will on either OS: Something this old and crusty and slow is going to be a bad experience on the web. You aren't going to minimize the badness by using an older OS that purports to use less resources because....
1) Older versions of Mac OS X generally don't consume less resources. Mac OS X will expand to fill the computer you put it in.
2) Most modern web pages are simply so huge and complicated that any of them will overwhelm an old Mac.
Remember, it has been a few years since smartphones became faster than the Quad G5, and the Quad is already over twice as fast as something like the dual-core 2.3, dual 2.5, or dual 2.7. Those are themselves something like four or six times faster, in total, than a G4 like this.