If all else fails, there's the cloud. But if you want something faster, say a home network and you have a mixed environment with Tiger, Lion, OS 9, and Linux Macs to share files on, you have several options to play with.
Target Disk Mode
If you don't mind rebooting and need to move large files quickly, Target Disk Mode is a good option. Target Disk Mode works on any Mac with native firewire. Just start up holding the "t" key, attach it to another Mac via firewire, and it will automount as an external hard drive on Macs running Mac OS 8.6 or later, and on any recent Linux distribution, too.
Another option is AFP (Apple Filing Protocol). This is the "Personal File Sharing" option in OS X's Sharing preferences. After Tiger it was changed to just "File Sharing," which is also the name of the corresponding control panel in OS 9. In Linux, you need to install the Netatalk package for AFP client and server capabilities, no post-install configuration necessary unless you want to do something fancy;)
For sharing between OS X machines, just activate the File Sharing option on the server side, then on the other Mac choose "Connect to Server..." from the Finder's Go menu, enter its router IP address (afp://xxx.xxx.x.xxx), followed by the server machine's administrator name and password. To connect a Linux machine, go to the network folder in your file manager and where you see your host's network icon, double click it and enter name and password. If you don't see the icon you can also enter the IP address in the text field.
When adding OS 9 to the mix, there are a few caveats. Apple has been deprecating legacy Appleshare and Apple Talk services like crazy, so a lot of the older stuff doesn't "just work." I can say that I can network with Tiger and Linux hosts using the Network Browser application in OS 9. And when using OS 9 as a server, make sure the TCP/IP box is checked in OS 9's File Sharing control panel. With Snow Leopard, I could only connect to an OS 9 server, not the other way around. It may take some trial and error to figure out what works on your setup, especially if you have Snow Leopard and up, but I'll leave you some links:
This System 7 Today link shows how to network System 7 and Leopard.
A 68k Macintosh Liberation Army thread goes through a lot about Lion and old Macs, and down thread has a hack to replace Lion's AppleFileServer.app with Leopard's to enable connecting with OS 9.
Also, Lion deprecated older AFP authentication methods causing connections to fail, so look here to re-enable them.
(UPDATE: And here's probably the most useful link of all, a very comprehensive post on Mac networking, including a handy chart showing your networking options depending on what Mac OS version you're on.)
There's also SSH and SFTP (SSH File Transfer Protocol). The server side is activated in OS X with the Remote Login option in the Sharing preferences. Linux systems usually have the client and server software automatically installed with the openssh-client and openssh-server packages. As far as clients go, Cyberduck is a great SFTP client for OS X (3.2.1 for Tiger available in their changelog section). There's Filezilla or bareFTP for Linux, or you can even connect through a file manager. And there's an OS 9 SSH client called Nifty Telnet SSH.
There's also ftp, but that's too boring to write about, har. And if you're transfering files from OS 9, check out my post about resource forks.