Print out your environment settings. Which variable may be used to store the CPU type of your machine?
Make a script that can say something on the lines of "hello, world." Give it appropriate permissions so it can be run. Test your script.
Create a directory in your home directory and move the script to the new directory. Permanently add this new directory to your search path. Test that the script can be executed without giving a path to its actual location.
Create subdirectories in your home directory to store various files, for instance a directory music to keep audio files, a directory documents for your notes, and so on. And use them!
Create a personalized prompt.
Display limits on resource usage. Can you change them?
Try to read compressed man pages without decompressing them first.
Make an alias lll which actually executes ls -la.
Why does the command tail testfile > testfile not work?
Mount a data CD, such as your Linux installation CD, and have a look around. Don't forget to unmount when you don't need it anymore.
Try all the mouse buttons in different regions (terminal, background, task bar).
Explore the menus.
Customize your terminal window.
Use the mouse buttons to copy and paste text from one terminal to another.
Find out how to configure your window manager; try different workspaces (virtual screens).
Add an applet, such as a load monitor, to the task bar.
Apply a different theme.
Enable the so-called sloppy focus.
Switch to a different window manager.
Log out and select a different session type, like KDE if you were using Gnome before. Repeat the previous steps.
Find an audio CD or a DVD or such, and try to play it.