ESP8266 - AT Command for Network Requiring Username AND Password
We can't tell without more information but it sounds rather like you will need a [WiFi_SSID, WiFi_PASSWORD] pair for the ESP to connect to the WiFi, and a [Network_USERNAME, Network_USERPASSWORD] pair for their network to let you use the network (as if you were logging onto it by hand from a terminal).

If so, you'd provide the first pair as you do now, with the AT+CWJAP... command, to get your device connected to the WiFi. For the second pair, your Arduino will probably need to interact with a user logon screen to provide the second pair. You'll need to get those details from the school's IT department, and possibly mimic a terminal.

Update:
As above, you need to find out whether the network requires 2 sets of credentials - the SSID and WPA2 key and network username and password. If it requires the latter, then your ESP can get you onto the WiFi but your Arduino will then have to mimic a user sitting at a terminal - read the network ID and Password challenges and provide the appropriate responses. Without know that, you can't know how to write the login code, nor can we. It sounds as if there are people on site who have already done what you're trying to do. They and their experience can be far more helpful than we, who have no knowledge of the school network.