1. First you need some unallocated space to create the partitions for each mountpoint (/var, /home, /tmp). Use Gparted for this.

2. Then you need to create the filesystems for those partitions (can be done with Gparted too) or use:

mkfs.ext4 /dev/sdaX

for example to create a new ext4 filesystem on the /dev/sdaX device (replace /dev/sdaX with your own device)

3. Mount the new filesystem under /mnt

 

mkdir /mnt/var
 mount /dev/sdaX /mnt/var

4. Go to single-user mode so that there is no rw activity on the directory during the process

 

init 1    /* for older ubuntu*/
systemctl isolate multi-user.target   /* for systemd */5. Enter your root password.

6. Backup data in var only (not the /var directory itself)

cd /var
 cp -ax * /mnt/var

7. Rename the /var directory after your data has been transferred successfully.

cd /
 mv /var /var.old

8. Make the new var directory

mkdir /var

9. Unmount the new partition.

umount /dev/sdaX

10. Remount it as /var

mount /dev/sdaX /var

11. Edit /etc/fstab file to include the new partition, with /var being the mount point, so that it will be automatically mounted at boot.

/dev/sdaX /var ext4 defaults 0 0

Repeat all these steps for /home and /tmp.