Basically all you need to do is download the installer and save it somewhere where you can find it again..
http://download.cnet.com/Avast-Free-...-10019223.html
So long as you don't run it on the old machine it won't affect your existing installation.
Just copy that downloaded file onto the usb stick or whatever you'll use to transfer it across, copy it onto the new pc and run it from there.
I'd recommend on the win7 machine to right-click on the installer and 'run as administrator' - it's easiest to do this if you have copied the installer file onto your desktop from the usb stick.