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.

USB 3.1 Gen2 is hard to beat. 10Gbps. But for HDDs the HDDs themselves will be the bottleneck. For sustained file transfers, perhaps no more than up to 3Gbps, even if installed in the same PC. Good SATA SSDs may manage closer to 6Gbps. NVMe SSDs are likely to saturate USB 3.1 Gen2. Then PCI-e directly between installed drives is the fastest. Normal cabled network is only 1Gbps.

File transfers to/from external multibay USB DAS can be done in parallel, involving multiple HDDs, and come close to saturating 10Gbps USB3.1 Gen2 speeds.

Then there is thunderbolt.

Start a rsync transfer when you go to bed. In the morning it will most likely be done.

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 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.

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

This will depend on the relative placement of these PCs

You could go over a wired network.

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.

I just transferred 12tb from one hdd to another via usb 3.0. Went at roughly ~180mbps and took 1.5 days

robocopy or rsync if you can’t place the drive in the target or an external enclosure.

External usb hdd

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