Surfaces on which not every pair of points is connected by a geodesic Let $S$ be a surface in $\mathbb{R}^3$.
I believe that, if $S$ is smooth, bounded, and closed, then,
for every pair of points $x,y \in S$, there is at least one geodesic $\gamma$
connecting $x$ to $y$.
This is because there is certainly a path between $x$ and $y$, and then one
could shorten it until it is locally shortest and is then a geodesic.

Q1. Is this correct? 

If so:

Q2. What are examples of
  surfaces $S$—perhaps not smooth, not bounded, and/or not closed— for which
  not every pair of points is connected by a geodesic?

 A: Any smooth surface in $\mathbb R^3$ that is a closed subset of $\mathbb R^3$ is complete as a metric space. Thus, if it is also connected, the Hopf-Rinow theorem guarantees that it is geodesically complete, which in turn implies that any two points can be connected by a minimizing geodesic. 
Therefore, the only ways that geodesic completeness can fail are


*

*if the surface is not connected, as in @Will's example, or

*if the surface is not closed, as in the case of a surface with a point removed.


This means that for connected surfaces, in a certain sense the only way for geodesic completeness to fail is if one or more points are "missing."  More precisely, if $S$ is connected but not geodesically complete, then it's not closed, which implies that there's a sequence of points $p_n\in S$ that converges to a point $q\in \mathbb R^3\smallsetminus S$. You can think of $q$ as a point that "should be" in $S$ but isn't. (But this doesn't imply that there was originally a smooth surface from which $q$ was removed -- for example, $S$ could be the cone $x^2 + y^2 = z^2$ with the origin removed.)
