A vector space of the form $\mathbb{F}_p^n$, where $\mathbb{F}_p$ is the finite field of prime order $p$, can be endowed with an incidence structure, i.e. a set of points (here just $\mathbb{F}_p^n$ itself ), a set of lines, and a set of relations between these points and lines.
A common incidence structure is the affine plane, where the three sets above have to satisfy certain axioms which resemble Euclidean geometry.
Of course there exist other incidence structures that are more general or exotic. I'd like to learn more about those, particularly when applied to $\mathbb{F}_p^n$.
My question is whether someone can give me tips to guide my search, as the literature on this topic is pretty intimidating for an outsider.
Finite projective geometries and finite polar spaces (including classical generalized quadrangles) are two common incidence structures based on vector spaces over finite fields, though their "points" are the one-dimensional subspaces rather than the vectors (and "lines" are usually the two-dimensional subspaces). Incidence can be the naturally induced containment, or defined using a sesquilinear or quadratic form.
Hirschfeld is the encyclopedic handbook (3 volumes), though not easy for beginners (or anyone).
There are numerous books and lecture notes you can find in pdf form from Peter Cameron's webpage (in particular, "Projective and Polar Spaces" and "Finite Geometry and Coding Theory"; also one on "Finite Geometry and Strongly Regular Graphs"). https://cameroncounts.wordpress.com/lecture-notes/
A nice introductory book by Simeon Ball called "An Introduction to Finite Geometry".
A very useful page with lots of resources: http://finitegeometry.org
A nice roundup done by a friend of mine when he was beginning his graduate program: https://anuragbishnoi.wordpress.com/2014/09/02/learning-finite-geometry-2/