how do I transfer files from Snow Leopard to Big Sur

As for the adapters, most Snow Leopard Macs have some variety of Firewire. Those that don't, do not give as easy access to data transfer over target disk mode.

However, all Mac OS X compatible Macs have ethernet, or adaptability to thunderbolt 2 to ethernet or USB-C to ethernet. You can setup Apple menu -> System Preferences -> Sharing and file sharing, and specify the sharing on the folders.

• https://www.dummies.com/computers/macs/how-to-enable-file-sharing-on-mac-os-x-snow-leopard/

Use the Go menu -> Connect to server on the Big Sur's Finder to connect to the network path specified in the Sharing panel.

You have a several options.

You can transfer the files to a USB stick or a USB hard drive, then connect that drive to the Mac running Big Sur.

Another option is to enable File Sharing on the older Snow Leopard Mac and have the Big Sur Mac connect to the older Mac.

https://support.apple.com/guide/mac-help/set-up-file-sharing-on-mac-mh17131/mac

Or you can put the Snow Leopard Mac into Target Disk Mode and using the appropriate cables & adapters connect it to the Big Sur Mac where the old Mac will appear as an external drive.