What's a fast way to copy a lot of files from an internal hard-drive to external (USB) storage?

Make sure that you are using a USB2.0 port and make sure the USB 2.0 controller says "high speed" -- a lot of disreputable manufacturers sell the "full speed (12)" USB devices with a prominent "USB 2.0" label on them, which is technically accurate but fools people who think USB 2.0 implies "High speed".

Also check if your USB hard disk has the "sync" mount option enabled? This is another cause for slowdown. You can remount the filesystem with

mount -o remount,async... /dev/usbdisk ...

Whatever is your interface to the harddisk you should use rsync to copy the data, it can resume transfers and files individually (--partial) , it gives you progress and checksums are verified on destination media.

In short:

rsync -avP src/ dst/

If you are transferring through a network interface add the -C argument to enable compression, in most cases you will be bandwith-bounded so it will not hurt performances even if the content is already compressed.

If you can you should tar your data before transferring it, it will relieve the filesystem of having to create many files, add timestamps, allocate space... on every file.
You'll likely see an improvement in speed of copy.

If I had a really huge amount of data to copy, I'd whip the drive out of the external enclosure and put it internally in the computer, and put it back in the enclosure when I was done. I keep a couple of SATA cables and drive bays open for just that sort of need. Opening up the enclosure and reclosing it afterwards it time consuming, but the copy itself will be miles faster.