There are a few different ways to transfer data between two computers. I believe the Snow (white) G3 iMacs had built-in FireWire ports so this should be fairly simple, just make sure you have the proper FireWire cables.
Different ways you can transfer files between Macs:
FireWire Target Disk Mode*
(more info)
Migration Assistant*
(more info)
Setup the Macs on a network and enable File Sharing
Burning files onto a CD
Using a USB disk drive or a Zip Drive
*Requires a FireWire port and a 6-pin to 6-pin FireWire cable
What kind of files do you need to transfer? If it's just some documents and other files you can simply transfer the data to the newer computer. If you need to transfer application preferences and data such as e-mail application data, or address book data you can use the 'Migration Assistant' mentioned above. The easiest thing to do is use the "Migration Assistant" program, which is located under 'Macintosh HD > Applications > Utilities' which will transfer most, if not all, of your G3 iMac files to the new G4 iMac.
If you need more help, maybe you can tell us what types of files you need to be transfered so we can have more detailed responses. Good luck