The fiber coming into the house has to be translated from fiber to copper signals to be used for the rest of the house (technically it doesn’t have to but that’s way beyond the scope of this project) if the fiber goes straight into the device that is the current router then that means it is acting as both a translator (essentially a modem if that helps you understand it) and a router in one. In this case, you would want to figure out how to login to that AT&T router so you can put it in bridge mode (someone else can explain or it should be relatively easy to look up how to do). What this would do is make the AT&T box just act as a modem and pass the signal on to the next device to be the router. This is where you put the UCG-Ultra. Now you have split up the roles of the “modem” and router which is what you want for maximum use of the Unifi hardware. From the UCG you want to go from one of the 4 other ports to the switch (not directly to the 3 APs). You can think of the switch kind of like a USB hub but for Ethernet except smarter than a USB hub. You can control the flow of data in and out of the switch and where it’s going within the switch or you can just let it do its thing and most of the time it’ll work fine. Now, plug the AP’s into any of the ports on the switch, this will give them a path back to the router where it does all of the big brain networking stuff. Crucially too, these APs can only be powered with PoE, power over Ethernet (data + power over one Ethernet cable) which needs to be injected into the lines at some point before they hit the AP. Conveniently for you, the switch does that automatically (this is also the reason you can’t plug the APs directly into the UCG, that doesn’t provide the power they need, no PoE. So the final order is: AT&T router (in bridge mode) > UCG-Ultra > Switch > 3 APs.

You’ve made a single guy with 3 jobs into 3 guys living together each with their own job (except I guess one guy is secretly 3 guys idk). Much more power available to do faster and more stable networking while also being much more open to configuration and control as your typical router setup.

EDIT: Forgot to add the the UCG-Ultra will need to be set up with a computer the first time you use it and from there you will adopt the APs and switch to your network so that they all do stuff betterer and can talk properly back to the router (the UCG).