Tarski's axioms formalize Euclidean geometry in a first-order theory where the variables range over the points of the space and the primitive notions are betweenness $Bxyz$ (meaning $y$ is on the line segment between $x$ and $z$, inclusive) and congruence $xy\equiv zw$ (meaning the line segment between $x$ and $y$ is the same length as the line segment between $z$ and $w$). One of the features of this system that I like is that it is somewhat modular; the axioms of upper and lower dimension can be chosen independently of the other axioms to fix the dimension of the space, and, notably, the axiom of Euclid can be simply replaced by it's negation to change it into an axiomatization of hyperbolic geometry.
It is very easy to adapt Tarski's axioms of Euclidean geometry to axioms of hyperbolic geometry. I am wondering if something similar can be done for spherical and elliptic geometry. From what I can tell, the answer is yes, but the betweenness and congruence relations have to be changed because of the different topology of spherical and elliptic geometry, compared to Euclidean and hyperbolic geometry.
Some more details
Specifically, the relations in Tarski's axioms indirectly rely on the fact that two points uniquely define a line segment in Euclidean and hyperbolic geometry, but this is not the case in spherical and elliptic geometries. Generally, two points will define two line segments, one going around the sphere the short way, and the other the long way. (In elliptic geometry, the line which is made up of these two line segments is still unique even when the points are maximally distant. In spherical geometry the problem is worse as the line is not uniquely defined when the points are antipodal.)
So I think that the betweenness relation would have to be modified to a relation meaning something like "$x$ is on the line segment that goes from $y$ to $z$ through $w$", and the congruence relation would have to be modified to something meaning "the line segment from $x$ to $y$ through $z$ has the same length as the line segment from $u$ to $v$ through $w$" - every line segment specification requires an additional point of information. But I am having difficulty figuring out how the behaviour of these new relations would be axiomatized. (And they very well might be different between spherical and elliptic.) Does anyone have any ideas?
Once the basic properties of these relations were axiomatized, I do not think it would be too difficult to translate the various geometrical axioms of Tarski's system (segment construction, Pasch's axiom, five-segment axiom, dimension axioms, and an axiom making the space curved). But the trick is getting the primitive relations to work correctly.