Use the following command to backup your home directory:
cd ~
tar -cvf /media/user/drive/home_backup.tar .
The first command ensures you are in your home directory. The second command backs up everything in this directory to an archive file called home_backup.tar
on external drive mounted under /media/user/drive
(you must adjust that part to reflect the actual path where your external drive is mounted).
On the new machine, conversely, use the following to restore your home directory:
cd ~
tar -xvf /media/user/drive/home_backup.tar
Try to avoid writing anyhting in the home directory while backup/restore is in progress, otherwise contents of the files that are written may become corrupted, either in the backup copy or in the restored file. (Be aware that some applications, like web browsers, are heavily writing to home directory, so it's best to avoid using any applications during backup/restore).
It is recommended to logout from GUI and login again after restore is completed (you are also restoring GNOME configuration files this way, some of them might get overwritten).
As @ubfan1 noted in the comment, ensure that your user ID and group ID on the new machine is the same as on the old one before attempting restore.