Changing user permissions
 

I've added an user in my Slackware...but now I'd like to modify his priviledges in the system, for example access to mounted disks, shutdown command etc.
How can I do that?

there's a couple different things here
1) access to mounted shares
-- you grant these through /etc/fstab, and options like gid/uid/users/umask

2) shutdown command
-- you grant this by giving the user sudo power... read the manpages for sudo, visudo, and vi (the editor used in visudo)

If you are doing specific file(s), use the chmod command.
[Other useful related command are chgrp and chown.]

Sheng-Chieh