You might want to take a look at Liam’s thread on hacking electronics, it goes over how to “push” a button with a microcontroller:
The solenoid route is a valid one, but solenoids require a bit more effort to drive:
As for the code, what you’re looking to do isn’t too hard, it was actually my first project that I did with an ESP8266 (now replaced by the ESP32) It pinged google then fired a relay to turn off the DC supply to our router. I’d look into the basics of Python and GPIO on a Pi, and ways to check for network connectivity
