If copying to drives on the same computer, the other drives must be writeable (such as FAT). Then you need to mount the partition as writeable (Knoppix defaults to read-only as a safety measure). See answer #6 for details. Then you should be able to drag and drop the files to the new disk (or use the command line interface, although it sounds like you are looking for drag and drop).

If you want to copy the files to a different computer on a network, there are many ways to do this type of transfer. Persoanlly, I like to run an FTP server on the receiving system and use Konquror as a FTP client on the Knoppix system (it is significantly faster than dealing with Microsoft file sharing, and my personal observation is that MS file sharing seems to deliberately drag it's feet when it detects a Linux system). See answer #5 for more details.