Suppose $T$ is a complete first order theory with a finite axiomatization. Must $T$ be $\aleph_0$-categorical? If not are there any simple examples of finitely axiomatized complete first order theories which are not $\aleph_0$-categorical?
The answer is no: for a simple example, take $Th(\mathbb Z,<)$. The axioms are:
Getting examples which are lower in the classification (e.g. stable) is much harder, and this is the point of the exercise in Marker's book which Garrett mentioned in the comments above.