Well, to transfer files you can mount the Ubuntu root partition (or
/home if it is on a separate partition) and just copy the files you can access on your USB that way.
Let's say you have a directory on your USB called Documents that you want to copy to your installed Ubuntu's home directory. You could do this...
First, from your full installation, find out what your root partition is on your installation by doing
Take note of the partition mounted on
/home if it's separate)
Now boot the live usb and mount that partition. Replace
/dev/sdxY with the correct partition name
sudo mount /dev/sdxY /mnt
Now your home directory is located at /mnt/home/username so you can do (replace with your real username)
sudo mkdir /mnt/home/username/casperdocs
sudo cp -r Documents /mnt/home/username/casperdocs
to copy the Documents directory to
~/casperdocs on your installation
Very important - don't forget to unmount the partition when you are done:
sudo umount /dev/sdxY
now reboot into your installation and you will find the files.
With programs (assuming you mean repository packages) you should install through
apt rather than trying to copy their files this way.