The title pretty much describes it. So, should I connect my older, weaker modem (which in its own can broadcast a wifi signals) to a new router that can broadcast much better through an Ethernet cable? If so, would the signals get better or would it stay the same?
Router - directs traffic (aka “routes”) and enforces access rules between different devices on the local network, and (if it’s also a “gateway router”) between devices on the local network and the internet
Access Point (“AP”) - creates a WiFi “hotspot” that’s connected to the local network (and from there, through a gateway router to the internet usually).
Modem - converts and transmits network traffic between two different physical formats: e.g. between fiber optic and ethernet, between cable internet and ethernet, between DSL and ethernet, etc.
The device that you get from your internet provider often has all three functions built in. Confusingly, many people refer to it as just a “router” or “modem” which doesn’t accurately describe it.
I would disable the WiFi in the cable modem and add an AP, or better yet, buy your own modem, router and AP and return the cable company device (and stop paying the monthly rental fee).