This is fairly simple. You will want to do a complete DB export from mySQL and then import that DB into the new installation's DB. The filestore can just be copied straight to the new installation. You will need to make sure that your file permissions are correct on the target after moving the filestore.
A couple of points though. Note that this assumes you are installing to the same version of RS. If your filestore and DB is large, you may want to do this from the mySQL command line if you have access to that. If not, (and it's large) you may want to do the import in stages. If you're not familiar with mySQL, you may want to have someone experienced with it do that for you. Still, I'd try this first. If you have phpMyAdmin installed, you can simply go to the EXPORT menu for the entire database . Select "Custom - display all possible options" in the Export Method. In Output, choose "save to file". For format, choose "SQL". Under "Format-specific options" choose "Dump table / data" (you only want to export the data not the structure). Save the file, login to phpMyAdmin on the new install and use the IMPORT function which should be pretty obvious. Then copy the filestore. Obviously, you need to ensure your config.php is setup/changed for the new DB server on the target.
Hope this helps; feel free to ask more specific questions and I'll try to help.
-Arne