I have a new way of looking at a router thanks to Alex. I understand the concept of public and private ips but I always looked at it as network facing and internet facing.
Viewing them as separate halves I think is more fitting like 2 sides of a coin. It tends to remind you of the wall that divides the LAN from another LAN in a different Domain, WAN and/or Internet. The router acts as a gateway through that wall.
The basic building blocks of LAN also helps in visualizing how they are connected and networked. Practicing with the cmd command lines serves as further reinforcement of what one knows.