While your G5 may be outdated compared to newer Macs, the internet is still your friend and very accessible. Thanks to great open source projects and other utilities, the web is wide open. You'll find recommended web browsers and other applications below. For apps that are free, consider supporting them with donations or notes of gratitude to keep them available for the foreseeable future for our G5 machines.
Last Updated: 4/23/2016
My recommended browser for G5s is TenFourFox. While the default icon may seem a little off-putting, this browser zips along, doing its best to remain current alongside major FireFox releases despite significant technical challenges. Fast and flexible, the browser is definitely the most secure option available for PowerPC computers and features a host of great extensions that increase security and functionality. The developer is very responsive and helpful, addressing issues and spending tons of time rewriting code to keep our PowerPC machines alive on the net. Follow his development blog to find out how much work he puts in to this great piece of software for Power Macs.
A noted poster on the MacRumors PowerPC discussion forum, eyoungren recommends a series of preference tweaks to optimize performance for Power Mac users of TenFourFox. You can download a preference file containing the optimizations here. While I can't say that I notice a night and day type performance, some of the tweaks may help on certain configurations.
Leopard-Webkit is another great web browser for PowerPC machines, relying on ongoing updates from the WebKit project itself. A bonus to this package is that you can run some bash scripts to replace the standard Leopard WebKit framework with this updated WebKit engine system wide. (Of course, use at your own risk.) A very solid web browser that continues to be updated regularly - thanks to Tobias.
BTW, the 600+ builds do include some nice security updates to make it more secure and less reliant on old standards.
There are alternative browsers for our Macs, but only TenFourFox and Leopard-Webkit seemed to be maintained. You may read of other options, like OmniWeb, iCab, and Stainless, but only Roccat (link below) is somewhat current (as of late 2015). Older browsers simply cannot be recommended due to potential security holes.
For all of its issues and age, Mail.app remains my standard email client of choice. It still works well with Gmail (for the most part) and other email hosting services. Beyond it, your next best option is to go entirely web-based, using TenFourFox to access your email.
If you do continue to use Mail.app on Leopard, you might want to read this post on System Folder for potential quirks that exist when using services like Gmail.
In the spirit of TenFourFox, TenFourBird tries to keep the now ended Thunderbird email app available for use for PowerPC users. Once upon a time, I used Thunderbird regularly as an email client, but I found ultimately it didn't offer much over Mail.app. You might find it, however, useful and frequently updated. (But look at that awful icon.)
Letterbox is a nice, free plugin for Mail.app that adds support for a three column view in Mail.app. I love it, especially if you have a wide screen monitor, the extra width makes the app's use of space all the more efficient.
CyberDuck is a free file transfer client with support for stuff like FTP, SFTP, S3, WEBDAV, SSH, and more. It has a ton of features, a really well put together app. You can find it useful to connect to something like Box.net and other file sharing services and allows you to edit html/css/text files on the fly. The last version to work for Leopard is located here, version 4.3.1.
Espresso is a great app for web development, letting you punch in html and css code and see how it looks instantly through a built-in web preview. However, the app is no longer available for PowerPC machines. I run version 1.1.2, which I cannot find for download at the moment. This site is entirely coded with Espresso. If I do find an archive, I'll share it here.
Text Wrangler is a powerful text editor for the Mac. And it's free, which is always good. Made by Barebones Software, it's fast, lean, and very flexible. It hooks in with CyberDuck as well for quick and easy editing of html and css files on the fly. The last version for Leopard can be grabbed directly here.
LimeChat is a solid, minimal IRC client for Mac. Version 2.2.6 is the last version to run on Leopard. The interface is a little different, but it is very easy to customize and find other themes to make it look gorgeous as you chat.
Adium is the last refuge of instant messengers. I have version 1.4.5 as the last version for PowerPC, but the website suggests I download 1.3.10. Meanwhile, does anyone even do instant messaging anymore?
The Unarchiver is a great replacement for Leopard's built-in archive utility. You can unpack just about any file downloaded over the internet, even more obscure compression packages. Version 3.9.1 is the latest for Leopard. Again, it's free. You'll want to download the legacy version of the app from that page above.
Transmission (version 2.22) is the recommended version of this popular BitTorrent client for Mac. It's great software with tons of features. However, like all things PowerPC, it hasn't been updated recently.
BitTorrent is the official BitTorrent client. The latest version for PowerPC is 7.3.5. You may find it has more features or stability than Transmission, so give both a shot.
Sadly, Twitter has changed to the point that existing PowerPC compatible apps for Mac cannot login or display their feeds. I have not been able to find a working client - all of my old ones long since have stopped working or are incompatible. Click the link above to use the mobile twitter site, which is a reasonable work around.
Here are some twitter workarounds:
Chicken (formerly Chicken of the VNC) is a great VNC client for Mac that lets you use remote desktop features of Mac OS X and other operating systems. It is still maintained. The latest version is 2.2b2. Grab it and enjoy.