Transferring External Hard drive data to new External Hard drive

I thought I understood the process of connecting Ext HDs to my MacBook Pro and, under Utilities/Disk Utility being able to see the respective drives. I eventually wish to transfer data between 2 Ext HDs - the current one and a new one yet to be purchased.

To start with, I have connected my MyHome to the MBP by usb cable but nothing relevant appears in Finder.

And, when I select Disc Utility I'm getting the window shown below - which doesn't look right.

What am I doing wrong?