The 3DS is fully charged. When I press the power button the blue light turns on for a moment, and it makes a noise like it...

If your phone is connecting to the wifi but doesn't maintain the connection then the problem is likely with the server or router and not with your phone. Verify that the wifi works on a different network to rule out the possibility that your phone is the cause of the problem. If the wifi connection provided by your school is spotty, I would recommend just using your data instead.