To transfer your Firefox bookmarks you simply need to export them to an HTML file and then import that file to Firefox on your new laptop. My method below assumes that you've got a USB memory stick available but any other method of moving the the HYML file (such as emailing it to yourself from your old laptop and retrieving the mail from your new laptop) will do.
Plug a memory stick into a USB port on your old laptop. Open Firefox and go to Bookmarks > Show All Bookmarks. Click on Import & Backup > Export Bookmarks to HTML. Navigate to your memory stick's location. (You can choose your own name for the backup if you want to but the default 'bookmarks' is fine). Click 'Save'.
Then eject the memory stick and insert it into a USB port on your new laptop. Open Firefox and go to Bookmarks > Show All Bookmarks. Click on Import & Backup > Import Bookmarks from HTML. Navigate to the location of your 'bookmarks' file and double-click on it. Job done.