In short:
Make sure that the menu bar (File, Edit, View, etc) is visible in Firefox on both the 'old' and 'new' computers. If it's not, right-click at the top of the screen and put a tick next to 'Menu bar'
On the 'old' computer, plug in a USB memory stick (or SD card, or whatever).
Go to Bookmarks > Show All Bookmarks
Then go to Import & Backup > Export to HTML
Save the file to your memory stick
Eject the memory stick from the 'old' computer and insert it into the 'new' one'
Follow the same process as before on the 'new' computer but (obviously) choosing the 'Import from HTML' option instead.
Navigate to the location of your file and double-click on it.
Job done.
The above is the 'official' way to do it and it usually works well. However if you've got many thousands of bookmarks stored it can sometimes fail to import all of them. An alternative is to search for the file 'places.sqlite' on the 'old' computer, copy it to a USB stick and then replace 'places.sqlite' on the 'new' one with the old file. Dead easy and guaranteed to work!