answering the question directly...
To move a user from one computer to another, transfer all the files underneath the user's home directory (let's say the user's name is 'Mary' and their username is also 'mary') to the new home directory on the new computer.
In the Terminal of your old computer:
sudo tar cvzf mary.tgz mary
This will create a mary.tgz file. Copy this file to a USB stick.
cp mary.tgz /path/to/usbstick
While logged into your administrator user on your new computer, if you haven't done this already, go into System Settings -> User Accounts, unlock it, and click the "+" (in the lower righthand corner) to create a user 'Mary' (with username 'mary')
sudo tar xvzf /path/to/usbstick/mary.tgz
log out of your administrator user and log into 'Mary' and verify all the files & preferences have been transferred before you clean up or get rid of your old computer.