Let's get you going the way you wanna go.
You'll be using file sharing. Plan a small vacation from the time you start till the time it will finish. A long coffee break might also work. In fact, if you work a standard 9-5, do it before you leave the house.
One problem you might have is that you've opted with the ethernet cross cable to merely move data. So that means there might be licensing or other necessary application components not in their proper place. It's going to require quite a bit of labor I'd think. Then there's the potential for permissions issues. You're going to move data from the G3 (assuming you are running an OS X build though you don't really say fer sure) and it's going to all be "branded" or "married" to the permission settings from the G3. Then you'll ask it to play nicely on the G5 without an intervention.
But, that's not what you're asking, is it. You just wanna know HOW to get it from the shanty to the mansion.
Connect the cross cable between the two Mac using the ethernet ports on each.
Please start the G5 first. Fire up the G3 after connecting the cable AND doing the following on the G5:
System Preferences | Sharing | Services
check the Personal File Sharing check box and file sharing should start up on it's on right after...
Note the START button turns to STOP and it reads just above it "Personal File Sharing On".
Make sure you have a computer name in the upper portion of the Sharing preference pane.
Note that once Fiel Sharing is fired up, at thebottom of the preference pane there's some help
"Other Macintosh users cab access your computer at afp://xxx.xxx.xx.xxx/
Start up the G3.
If it's running OS X (and again I'm peeved at you for not stating) in the Finder click the Go menu and drop to Connect to Server. In the server address area insert the afp://address
and you should be able to mount the other drive or at least soem folders from it this way. You may need to authenticate with username and password set in the G5 so keep it fresh in mind.
Hope this helps you brother.