Display network information for your workstation: IP address, routes, name servers.
Suppose no DNS is available. What would you do to reach your neighbour's machine without typing the IP address all the time?
How would you permanently store proxy information for a text mode browser such as links?
Which name servers handle the redhat.com domain?
Send an E-mail to your local account. Try two different ways to send and read it. How can you check that it really arrived?
Does your machine accept anonymous FTP connections? How do you use the ncftp program to authenticate with your user name and password?
Does your machine run a web server? If not, make it do so. Check the log files!
From your local workstation, display a graphical application, such as xclock on your neighbour's screen. The necessary accounts will have to be set up. Use a secure connection!
Set up SSH keys so you can connect to your neighbour's machine without having to enter a password.
Make a backup copy of your home directory in /var/tmp on your neighbour's "backup server," using scp. Archive and compress before starting the data transfer! Connect to the remote host using ssh, unpack the backup, and put one file back on the original machine using sftp.
Make a list of open (listening) ports on your machine.
Supposing you want to run a webserver. Which services would you deactivate? How would you do that?
Install available updates.
How can you see who connected to your system?
Make a repetitive job that reminds you to change your password every month, and preferably the root password as well.