There's no need to restrict this conversation to H^2.

If a manifold with a metric (a.k.a Riemannian manifold) is connected, then any two of its points can be joined by a curve - this won't necessarily be a geodesic.

Geodesics are defined as "curves of least length", say like a great circle on a sphere. This is not so precise, as there are great circles that do not minimize length. What actually goes, is that on a geodesic, the tangent vector can be transported parrallel to itself - and this condition is given by a system of differential equations.

This explains the local nature of geodesy - Using standard ODE arguments. For given initial conditions, such as a point on the manifold and a direction on the tangent space, the system can be solved locally (i.e. there is always a geodesic between that point and the ones sufficiently close to it) and the solution is unique.

Hope this clarifies things a bit.