This may go beyond what you are asking for, but there is a wonderful book called Introduction to the Foundations of Mathematics by Raymond L. Wilder. I provided its axioms and an example of how they could be satisfied using SET cards in MESE 2528.
Here is again a list of the axioms:
The full book is available online (open access) and I have derived quite some joy over the years from proving the various theorems that follow the above-mentioned axioms. Wilder himself begins his discussion of these five axioms by showing that they can be "satisfied" with an interpretation around people as members of clubs, and remarks that it is achievable with just four people; so, he continues, this cannot be a full description of (e.g.) Euclidean geometry, since we know there are many more than four points in the plane!
The theorems that Wilder uses begin as innocuous; according to my own notes, the first one is:
Theorem 1 Every point is on two distinct lines
and become increasingly more difficult; e.g., given: $n$ total points; let $m$ be the number of lines containing a given point - this has already been shown as well-defined in an earlier theorem; and let $r$ be the number of points on a line - also shown well-defined in an earlier theorem. Then one has:
Theorem 10 The total number of lines is $m \cdot r$.
The book is from 1952 (the link above goes to its reprinted second edition from 1965) but I think it holds up quite well. Even if the full book turns out not to be ideal for your present purposes, I would recommend this as a reference; there are some nice problems and explanations scattered throughout.