How do I get client mode to work?

Well, using the Airport utility, you can go into manual mode and under the Wireless tab you can change between "Join a wireless network" where you can connect your computer to the wired port and use the Express as a wireless card with an ethernet cable. Or the "Participate in a WDS network" which will make your Express into a repeater for a current wireless connection and you will connect wirelessly to your Express which will be connected wirelessly to the school network.

To input your password you need to change the "Wireless Security" to the security that your school uses and input the username and password to connect to that wireless access point.

 

Hold on a second, are you trying to just connect your laptop to a network, or are you trying to connect the Airport Express. This, is the AirPort Express:http://www.apple.com/airportexpress/ The AirPort Utility, the program used to edit the Airport Express, is in the Applications/Utilities folder. You are configuring your laptops internet connection with the box you are showing. And I do see a spot to put in your username for the picture you have shown.

 

The Airport Express does not support using 802.1x authentication when joining an existing wireless network. You will not be able to use it in this way.