I need to give you three warnings before my suggested setup steps:
1. You must be connected to the Internet since you are visiting MacRumors, and I don't know what settings you are using for Networking or what kind of cabling from either computer. I don't want you to mess up your Internet access by changing settings or moving cables around, so before you change anything, make note of your cabling and how your control panel and preference settings are set now, so you can put everything back the way it was, if necessary.
2. Turning on file sharing when you are connected to the Internet can make your computers accessible from the outside world. Firewall software or hardware can avoid this by isolating your two computers from the Internet side of your connection. File sharing momentarily is a smaller risk than leaving file sharing on all the time, but it's a risk all the same. If you don't have the protection of a firewall, you might want to turn on file sharing only when you are disconnected from the Internet.
3. Since I'm writing this from memory without actually trying it myself, I hope I'm steering you correctly. It's late here in la-la-land (yawwwwwn), so I'll check this thread tomorrow in case you need more help. And, of course, I'm not the only one who will offer you help. Someone else may correct my information or help us both through followup steps.
Now, for my suggestions...
--- Which kind of connection ---
Ethernet is the usual approach. You only need to share one way (either "push" files from the G3 or "pull" files from the G4), but sharing will actually work both ways so you might as well set it up to do so. Of course, we only have to get one of the two directions working, so if one direction fails, you can still do what you want.
Although you can connect a Mac OS X computer to another computer as a firewire drive, you then can't use it directly because the Mac is simply pretending to be an external firewire disk drive.
--- Cabling ---
If both Ethernet ports are available, you can use a crossover Ethernet cable directly between the two machines, or run a normal (straight-through) Ethernet cable from each Mac to a hub.
In other words:
G3 <--crossover--> G4
or
G3 <--straight--> hub <--straight--> G4
--- Setup ---
On the G3:
In the File Sharing control panel, turn file sharing on.
Select the hard disk and choose File->Get Info->Sharing Setup. Click "Share this item".
In the Users & Groups control panel, open "Guest" and check "Allow guests to connect to this computer".
(Note: I don't have an OS 9 system here to check this. I'm telling you the OS 8 setup. I forget if OS 9 is the same.)
On the G4:
In the Sharing preference pane, turn on "Personal File Sharing". This will allow access to your Public folder, e.g., /Users/yourname/Public from other computers.
In the Network preference pane, AppleTalk tab, make AppleTalk active.
--- Networking ---
In the G3's TCP/IP control panel and the G4's Network preference pane, TCP/IP tab, you need settings that let the two computers see each other.
Typically...
If they are configured with the "Manual" method, you'd want them to have similar but different IP addresses. For example, if the Subnet Mask is 255.255.255.0, the two IP addresses should differ only in the fourth number.
If they are configured with the "DHCP" method, then both computers should be connected to your router, because they will be given unique addresses by the router or your Internet service provider.
If you say more about how you are currently set up (dial-up or Ethernet, and what Network settings), I can probably be more specific.
--- To access the other computer's files ---
On the G3:
Open the Chooser and choose AppleTalk. You should see the G4 show up on the right side shortly. Pick it and you should be able to connect as a guest and see the Public folder on the G4.
On the G4:
In the Finder, select Go->Connect to Server. You should see the G3 show up in the window shortly. Pick it and click Connect. You should be able to connect as a guest and see the hard disk.
In either case, you can copy files back and forth.
See? Nothing to it!