Most of the time, you don't need to move the installation. Just resize the partition.
http://gparted.sourceforge.net/livecd.php - this livecd is able to resize partitions.
If you really need to move to another partition, then rsync is your friend.
Read its manual if you need to, but my hints are to use the -a switch ('archive' like in cp command). Rsync is better than cp, because sometimes cp will fail, or you will need to stop it etc. No need to start from scratch afterwards
rsync -rvva /source/dir/ /dest/dir/
be sure to use slashes after the paths.
You have generally figured out most of the things you need to do. Other than these things, you will need to edit your /etc/fstab file, so that the new installation is able to mount the filesystems on the updated partition schema.