For all-in-one devices, a single unit handles wireless connections
(broadcast AP), hardline connections (switch), routing duties (router,
firewall, etc.). Most residential-grade routers are really router /
switch / access point. A few provider-issued devices are modem /
router / switch / access point. Your new setup will be shifting duties
to separate devices.
Modem - connects to internet lines coming into the building, often
provided by or leased from your ISP.
Router - Brain of the network, routes connections where they need to go.
Switch - Paths for the network to follow; your hardlines of CAT will
make their way to a switch, and then to the router.
Access point - provides a wireless signal that devices can connect to;
will also be connected to a switch.
As Joshua replied already, having everything on a single SSID
(wireless network name) is best. A "mesh" system, such as what Cisco,
Unifi, or certain residential devices have, handles the handoff
between APs as a device moves from one coverage area to another.
You could also, depending on your setup, direct students to one
wireless network, staff to a second, etc. Unifi can handle that
internally by creating VLAN (virtual local access networks), and
moving prespecificed devices to specific networks.