Rasta: Welcome to the forum. You have a couple of options (at least) here. You can go on line and update with your current version. I doubt much can happen since you have a relatively current version.
The way I do it when I want to add avast to another computer, is to download from the web site to disk the current version on the computer I use at home, and then install from the disk onto the new computer. When you go on-line you will be fully protected from the get-go.
Moving the files from your laptop should be a simple matter of moving files to CD, assuming you have a CD-RW. Then install as above on your new one from the disk you created.
I hope one of these ways seems do-able to you. If you need more help, please post again, there are plenty of us here to assist you.