Just one small point for clarification, I hope.
If you own a Desktop Mac (iMac), then the External Hard Drive (EHD) should remain mounted and Time Machine will do its duty, hourly. The first Backup (BU) will take a while, but subsequent BUs are incremental and will be much shorter. It does all this in the background.
But if you own a Notebook, then the situation is different. Sure, you mount the EHD and carry out your initial TM BU; and you can keep the EHD mounted until it's time to take the Notebook with you. Then you must unmount the EHD.
During the time the Notebook is on its travels with you, TM creates snapshots of the incremental BUs and store them temporarily on the Mac's HD. Then, when you get home, remount the EHD and the snapshots will automatically to transferred to the EHD; thus clearing the storage space on the Mac's HD.
Hope that makes some sense.
Ian