My eldest kid asked for a network rack for their birthday.

I gave them my slightly used 12U rack and my old 1G switch. That switch is not connected, yet.

I have a UI enterprise pro 24 port POE main switch. Will that 1G switch work correctly if its plugged into my POE switch or do I need an aggregation switch to connect the two switches?

It will be fine plugged into your switch.

Strange request for a kid, but pretty cool.  I'd oblige them. Is he/she/them going to a CCNA boot camp or something similar?
The St. Louis public tech schools have networking curriculum now, dang kids can graduate with a Diploma for HS and a CCNA or CCNP...

On your UI POE switch just make sure you can either shut POE off or that it will sense and shut it off.   I have a few switchs that when I turn on POE, it gives me a warning and says it will damage either the port or device plugged in, if not POE.  If it will or not is a different story.  Rather save a port and not loose one if I don't have too.

Also, sometime when switch to switch, you need to use a cross over cable vs a straight though unless they are auto-sensing that too.

It should, but may require some reconfiguration, depending on how each switch is configured now. If you're using all vlan1 and unmanaged switches, it should be plug and play.

Thank you all.

Why are you using a switch?
Put a router in instead, and give that lab some isolation from your home's /24.

Concord Aerospace Switches by FredMan, on Flickr

Oops, wrong kind of switch!

Why do that when he could just make a separate VLAN for the lab devices?

I trashed my wife router when I went with UI products. Currently I have a UI Dream Machine Pro and Enterprise 24 port POE switch. Wifi is provided by an access point.

Good idea about the vlan.