Ok, from the output you have posted,
You have /dev/sda, a 10.3Gb disk.
This has a 9.5Gb Linux partition /dev/sda1
which has the mount point / and is type ext4.
There is also a swap partition /dev/sda5 in an extended partition container /dev/sda2 of 475Mb.
You also have /dev/sdb, a 6.4Gb drive
which is not mounted, but has a 5.9Gb linux partition and a swap partition in an extended partition container (not organised very well, as the partitions are not aligned to the cylinders).
To use this disk, you need to mount it on to your existing filesystem. The main linux filesystem divisions are /boot, /home, /usr, /var. 6Gb is fairly large, so my suggestion would be to use the second disk for something like /home (all the user files). Others may make different suggestions.
The following should be carried out in the recovery console boot or in a single console session to prevent locked files (after boot, before logging in with gdm/kdm use ctrl-alt-F1 to switch to the console, log in as a user).
use mkdir to create /mnt/newhome
**sudo mkdir /mnt/newhome**
mount the second disk
**sudo mount /dev/sdb1 /mnt/newhome**
If you get a mount error it will almost certainly be because the partition has not been formatted as ext4. Use mkfs.ext4 to do this
**sudo mkfs.ext4 -j /dev/sdb1**
and retry the mount command above.
run a mount command to verify the mount
**sudo mount**
You should now see a reference to /dev/sdb1 in the list of mounted volumes.
You can cd to /mnt/newhome to check the new volume
**cd /mnt/newhome
ls -l**
If the directory is not empty, delete files
**sudo rm -Rf /mnt/newhome/***
cd back to /
**cd /**
Copy the files from /home to /mnt/newhome
**sudo cp -R -p /home /mnt/newhome**
Once this is completed, rename /home to /oldhome.
**sudo mv /home /oldhome**
create a new empty home directory
**sudo mkdir /home**
unmount /dev/sdb1
**sudo umount /dev/sdb1**
check permissions on /home
**ls -l /oldhome**
set /home to have the same permissions using chmod/chown
mount /home
**sudo mount /dev/sdb1 /home**
use ctrl-alt-F7 to switch back to a gui console
log in to ensure that /home is working.
In a terminal window type
**sudo gedit /etc/fstab**
copy the /dev/sda1 line to a line just below it, and edit /dev/sda1 to /dev/sdb1, and change / to /home (leave all other options the same). Save the file.
log out of the gui and switch back to the text console ctrl-alt-F1
unmount home
**sudo umount /dev/sdb1**
confirm with mount
mount /home using fstab
**sudo mount /home**
confirm mount
reboot
**sudo shutdown -r now**
at the login screen, ctrl-alt-F1 and login
check that the mount has occurred correctly
**sudo mount**
and that files are on /home
**cd /home
ls**
ctrl-d to log out, ctrl-alt-F7 to go back to gui
log in
at some point when you are confident that everything is working you can delete /oldhome.
To back out this change,
**sudo umount /home
sudo rm -R /home
sudo mv /oldhome /home**
and remove the /dev/sdb1 line from /etc/fstab.
Si
