if your using IE then open IE click on the file menu and select import and export, follow the wizard through selecting "export cookies" and save them somewhere you can find them again.
now it's upto you how you get the exported cookies onto the other computer. (copy over network, burn to CD, use a USB memory stick..... whatever works for you)
Once you have them on the new computer then repeat the "import export" thing but select import cookies this time, point it at the file you have just copied over and away you go.