Your set up the directory (folder) structure for your media files yourself. It is unique to your computer.
The structure should look something like this, for example.
C:
Users
Your PC Name
My Documents
My Music
J River Music Media Files
Artist 1
Album 1
Track 1
Track 2
...
Album 2
Track 1
Track 2
...
Artist 2
Album 3
Track 1
Track 2
...
Album 4
Track 1
Track 2
...
Where Artist, Album and Track are the actual names for your music files. The unique part in this example starts with J River Music Media Files. That is your top level directory for you media files. With explorer you can click on the J River Music Media Files folder and drag it over to the external dirve, something like K: Alternatively, you can click on that folder, hit Ctrl C, then click on the external drive (K:) and hit Ctrl V. You will then have the same directory structure on K:. You then move the drive to the new computer and use Windows Explorer to move the directory from K: to wherever you want it on the disk on the new computer.
Personally, I would not use the Microsoft Users/PC Name/My Documents/My Music structure, but would place the top level directory right under C: or whatever drive designation you have. But that is up to you.
If you want to use the Rename, Move Copy function of Library Tools, just select one of your views (e.g. Albums), click on the first album cover (it will become highlighted), scroll down to the bottom, and click on the last album while holding down Shift. All the albums should then be highlighted. Then right click, select Library Tools and Rename, Move Copy and then follow Bryan's instructions.