1) Copying files
You want to copy the FILES, not the whole partition ( including its free space ), so you don't need to resize the partition first. Boot from the livecd and mount both the HD and SSD ( after formatting a partition on the SSD of course ), then copy all of the files over:
sudo cp -Tax /media/hd /media/ssd
Use the correct names for the hd and ssd mount points of course. Then you just need to edit the /etc/fstab on the ssd to point to the new fs UUID ( you can look it up with
blkid ). Finally you need to install grub on the ssd.
2) Dealing with GRUB
a) Command line:
for f in sys dev proc ; do mount --bind /$f /media/ssd/$f ; done
Of course, use the correct device for /dev/ssd. The whole disk, not a partition number.
b) "Recommended repair" magic button in Boot-Repair:
sudo add-apt-repository ppa:yannubuntu/boot-repair
sudo apt-get update
sudo apt-get install -y boot-repair && boot-repair
Finally reboot and make sure your bios is set to boot from the SSD.