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Do you have a system, that you want to move (without changing anything else)?
Do you have the hardware where you want to run it?
0. It is important to backup your system before such an operation (backup to a separate external drive)
A - automatically
I think it will be easiest to simply copy the current /home (preserving the permissions and ownership) to a new partition (on the HDD to be used for it), and then install a fresh Ubuntu system with the root partition on the SSD and the home partition on this new partition.
Prepare the partitions before installing the new system. Then during installation at the partitioning page, select 'Something else' and select the proper partitions of root and home and the proper drive for grub (the drive for example /dev/sda, not a partition /dev/sda1)
B - manually
1. Perform the changes when booted from another drive, for example an Ubuntu install drive.
2. the root partition /
Let us assume the source is mounted on /mnt. Create a partition on the SSD and use
to copy the content of your present root partition except /home
sudo rsync -Hav --exclude=/mnt/home ...
If to a new drive: Create the partition to become your new /home and use
to copy the content of your present /home to it.
sudo rsync -Hav ...
If on the same drive: Let us assume it is mounted on /mnt. Keep it until you have copied the root file system to the SSD drive, then remove 'everything else' from /mnt
and finally move /mnt/home to /mnt.
3. You need to edit the file /etc/fstab to reflect the changes of the UUID of the root partition and add a new line to mount the /home partition, and maybe also a new swap partition. I recommend also that you add the following options to the root partition (because of the SSD hardware) discard,noatime in the fourth field (it often starts with defaults)