Flatbed scanners have mostly embraced open standards and are compatible as a plug-and-play device on Linux and other platform.
Plug your scanner into your new Linux computer and turn it on. Launch the Simple Scan application from the Activities Overview.
Click the Scan button to initiate a preview of whatever you have on the scanner. Click the Crop button to determine what area on the flatbed you wish to save, and then click the Save button to save a full-quality scan to your harddrive.
Scanning directly into a graphics program is also an option, utilizing the Gnu Image Manipulation Program (commonly known as GIMP). You can learn more about GIMP and lots of other great Linux applications in the Software section of this site.