Quickest -

1. Take the lid off the source PC put the HDD in it
2. Copy the stuff.
3. Pull it out
4. Put it the other PC,
5. Your done

Formatting the HDD my be optional.

Connect both computers over ethernet and rsync the data across. It will still take several hours over gigabit ethernet though.

1. Take the drive from PC 1 and attach to PC 2 => copy / paste

2. External drive

Parallel port would be faster than serial, but if you’re using serial I’d go with Z modem for the compression and error correction.

Over LAN with flavored solution of your choice.
Other than that, I use one of my spare drives and my USB docking station if I need to move it outside of my own house. I never really had a need for dedicated external drives but a docking station is handy for many things, this being one of them.

This will depend on the relative placement of these PCs

You could go over a wired network.

Fastest “normal” way would be write it to a USB 3 drive on machine 1, read from drive later on machine 2. An SSD would be best. Even with having to do the write and read in two phases, I’d still expect it to be faster than a 1G network.

(Less normal ways would be things like Thundrerbolt SSDs, 10G or 40G ethernet for direct transfer, etc. But if you have that kind of tech already in place, then you’re not asking. ;)

rsync over sshfs over Ethernet.
I transferred 10 Tb over night, but it depends on how many files do you have.

1 Tb shouldn’t be too bad. You could try an external HD, but that becomes a double operation (1 copy to the drive, 1 copy off) or use your home network.

Create a share of the folder(s) you want to copy and then access from the new PC to copy and paste to. Speed will depend on your network speed AND drive speeds.

What is the best method?

Sometimes the best method isn’t a transfer at all. Don’t forget transporting is an option.
Do you need to transfer the files? or could you just use an external you share between different PCs?
If an external works, but seems inconvenient, you can fix that by having the drive connected to a USB switch. When connected to a switch you can press a button or use a remote to change which PC the drive is connected to giving you instant access to the drive contents without having to sync/copy anything over.

Links below for examples of USB switches
https://www.amazon.com/Cable-Matters-Sharing-Computers-Peripherals/dp/B083JKDNRJ/
https://www.amazon.com/Sabrent-Computers-Peripherals-Indicators-USB-SW30/dp/B074TYDJK2/

Over a network.

Not WiFi though!

You could hook a single network cable between them. Give each machine an IP address, share the folder that you are copying to or copying from (your choice) and drag the files over. Or you could use RoboCopy to potentially do it faster.

Or just get a 2tb usb external drive and use that, it could be slower or perhaps faster than the network option depending on many factors but also more expensive as you have to buy the drive!

External usb hdd

I have a PC in a spot of my house I can’t get a cable to. But that’s the PC I do my Blu Ray ripping and encoding on, so I keep a 5tb external drive for moving the files to my NAS afterwards. Sometimes it would take days over WiFi instead of minutes to hours over USB 3 twice. (and piss off my wife for saturating the wifi, heh)

rsync over a wired network. That way, I can stop and resume at any time.