How to calculate/find the network ID of my home router and second hop router?

Question

Going by the book definition a router is connecting 2 or more networks depending on the number of interfaces inside of it. Let's discuss the simple case, my home router which is connecting LAN with WAN. I tried to traceroute multiple IPs and the result is the following: first hop is my public ip (assigned by ISP and listed on whatsmyip), then second and third hops are private IPs from the 10. range, then at fourth hop is a public IP and so on until destination. So i suppose my wan port is part of a private network inside my ISP and traffic is going out of my ISP network at 4th hop. My question is: how can i know the network id for my wan interface since it has a public ip attached and which is connected to a private ip at second hop in tracert? Is it possible to have a private network with devices having public IPs and private as well? For my LAN network i can figure it out based on ip and subnet mask id but I don't know how to do it when traffic flows out from my router. Or they are simply 2 routers directly connected by the cable without forming a network? I mean if 2 routers are connected shouldn't they be in the same subnet?! If so how can a public ip be in the same subnet with a private ip. It's confusing... Thank you!