On the axiomatic front, you might find the axiomatization of Desarguesian planes useful for your project. A very nice exposition is chapter 2 of Emil Artin's "Geometric Algebra", which I'll now summarize.
The undefined notions are points, lines, and an incidence relation between them. The axioms for an affine plane are
- Through any two points there is a unique line.
- Through any point not on a line there is a unique parallel line through that point.
- There exist three points not all on the same line.
Using these three axioms (the second really) one can show that being parallel is an equivalence relation on lines. One can then define an affine transformation $T$ to be one that preserves this relation, i.e. if $\ell_1\parallel \ell_2$, then $T\ell_1\parallel T\ell_2$.
More significant are the affine transformations that leave the parallelism class invariant, i.e. such that $\ell\parallel T\ell$. Artin calls such affine transformations dilatations. This is because dilatations are uniquely determined by where they send two points, as then the location of the third point is uniquely determined by the parallel postulate.
Consequently, he defines a translation to be a dilatation that has no fixed points, and a dilation to be a dilatation that has one fixed point. One can show that dilations with no fixed points are uniquely determined by where the send a single point. This allows us to visualize the action of translations on the plane as arrows: an arrow for a translation indicates that the translation takes the starting point to the ending point. It is instructive to prove that any two non-collinear arrows that come from a translation can be realized as opposite sides of a parallelogram. This is the affine version of the statement that translations are specified by the data of a magnitude and direction.
We do not yet know that any arrow comes from a translation, however. This is precisely the content of axiom 4a of Desarguesian' geometry.
4a. Desargues' theorem for parallel lines. Given three distinct parallel lines $\ell_1\parallel\ell_2\parallel\ell_3$, suppose we have points $A_i,B_i$ on $\ell_i$ so that $A_1A_2\parallel B_1B_2$, $A_2A_3\parallel B_2B_3$. Then $A_1A_3\parallel B_1B_3$.
The point is that if the arrow $\vec{A_2B_2}$ were to come from a translation, then that translation would have to send $A_1$ to $B_1$ and $A_3$ to $B_3$. But this is actually a dilatation if and only if $A_1A_3\parallel B_1B_3$, which is exactly what this version of Desargues' theorem asserts.
4b. Desargues' theorem for concurrent lines. Given three distinct concurrent lines $\ell_1,\ell_2,\ell_3$ with common point $O$, suppose we have points $A_i,B_i$ on $\ell_i$ so that $A_1A_2\parallel B_1B_2$, $A_2A_3\parallel B_2B_3$. Then $A_1A_3\parallel B_1B_3$.
This second version of Desargues' theorem says that giving the points $O,A_1,A_2$ does in fact specify a dilation fixing $O$ and sending $A_1$ to $A_2$.
An affine plane satisfying axioms 4a and 4b is called a Desarguesian plane. These are significant because the translations plus the identity dilatation form an abelian group, and conjugating these by a dilation results is an endomorphism of this abelian group that preserves the trace lines of the translations. As Artin describes in detail, together with choosing the three non-collinear points $O$, $A$, $B$ from Axiom 3, this establishes a correspondence between Desarguesian planes and $2$-dimensional modules over division rings. In other words, the 5 axioms above are all you need in order to relate synthetic geometry to coordinate geometry.
Artin does further cover the fact that requiring an order relation on the points on the line of a geometry corresponds to requiring that we have an ordered division ring, which ends up being an ordered field, so something inbetween $\mathbb Q$ and $\mathbb R$. What Artin doesn't do is discuss lengths, congruences, and similarities. I worked out some of this some time ago, but I'm not full satisfied with it. In any case, the relevant axioms for lengths and angles are
A. A translation-invariant distance function satisfying the triangle inequality (i.e. a norm on the vector space of translations).
B. The axiom that across any line there is a distance-preserving reflection, where a reflection is an involutive affine map fixing that line. This establishes the necessary relation of perpendicularity (equivalent to a positive definite inner product on the vector space of translations).
This may seem strange, but remember that the affine plane is still the affine plane even if you look at it at an angle. Then what used to be circles now look like ellipses, but all of geometry still holds. You can get rotations as composed reflections, and this is enough to recover traditional side-angle-side congruence and angle-angle-angle similarities.
P.S. Vectors are translations.
