Your local and public IP address are two different things.
What you're looking at is your local IP address, the Gateway address is your routers IP and typing that into a webbrowser can get you access to the router page. If you can guess or know the password/username then you can control the internet of your school and possibly shut it all down.
With a regular IP address you can't do much of anything besides try confirming someones identity or if you know of open ports on their network (backdooring) you can access or spam those.