Reading maps:

Find That Country: My brother and I had a world map with the names of all the countries listed underneath. We took turns challenging each other to find the most obscure-looking ones off the list on the map. Since at first neither of us knew just where Micronesia or Lichtenstein might be, the only way to find them was by a fairly exhaustive crawl over the map, which meant we picked up something of the place-relationships of other countries as we searched. A parent might choose to pick more readily findable countries and offer hints at need. As I remember my brother and I mostly played this one on our own, and it amused us and kept us out of our mother’s hair for extended periods of time… No country list is needed; kids can just look at a map or globe and find country names which they can then challenge each other to find.

Bordering:… In this game the asker sat facing the map and said “Name the countries bordering…” the country of their choice. (“Russia” reliably induced groaning as it had so many neighbors.) The guesser sat where they couldn’t see the map. The asker said “Yes” or “No” to each of the countries the guesser named, and let the guesser know when they’d found all the correct answers. The guesser worked from memory or else guessed wildly until they got a ‘yes” on a country they knew something about.
Both these games would work just as well with US states as with countries.
Making Maps:
Simple mapmaking helped to make the maps we read more real to us.
Sometimes my mother had me draw maps of actual spaces I knew: my bedroom, my house, my yard, a section of my road… For the room and the house it was easy to pace off or even measure out distances, and figuring out how to scale those measurements down and draw a scale map was good arithmetic practice. For larger areas I guessed as best I could.
I was also encouraged to make imaginary maps of various levels of complexity. I made a ‘road cloth” for my younger brother. We took an old bedsheet and fabric crayons, and I drew roads wide enough so that his toy cars would fit in the lanes. I added in parking lots and footprints for buildings; sometimes he built buildings on those sites with his blocks. I had to think a bit about how best to get from one place to another.
For my own amusement I drew elaborate maps of the countries I made up, complete with legends, obscure symbols and runes, and “Here Be Dragons” written in blank spaces toward the edges. Sometimes I made the maps to illustrate stories I had already started to write. Sometimes I started with the map and then began to have story ideas.