E.g. in moving from the G5 to a 13" MacBook Pro:

Connect the old Mac to the new Mac with a Firewire cable. Boot the old Mac in Target Disk Mode (Squiggly-T). Boot the new Mac.

The old Mac will show up as a disk drive. Copy over everything you want. (The more recent versions of MacOS will even transfer settings, accounts, etc., this way, if you want.) Shut down the old Mac. You're done.

In your case, since you can't boot the old Mac, I would remove the hard drive, and either install it in the new Mac as a slave or use an external USB/Firewire/eSATA enclosure or adapter to power it up through an interface of your choice. Then copy over everything that you want. Make sure all your critical software works from the newer drive. At that point, you can remove all your personal stuff then put the drive back in the old system. (There might be subtle differences in the video drivers, etc., in the old and new systems that might cause issues if you boot the new system with the old disk.)

Someone who is into Macs who is interested in a G5 is likely to reinstall stuff anyway, so you could also just keep the old drive and throw in a new/different one if they need it, as an alternative.

Just some options. Good luck.

Cheers,
Scott.