Go into your Library (Hard Drive/Users/YourUserName/Library), and open the folder called "Safari." There should be a file called "Bookmarks.plist." Transfer that file to the same folder on the new computer, and you should be done.
You can also drag bookmarks in and out of the bookmarks window and just transfer them as files. They will be .webloc files. Then just drag them into the bookmarks window on the new system.