Is energy always conserved?

If you all have trouble wrapping your heads around this, there is another way of explaining what is going on. I'm not sure if it'll be more illuminating, but let me try to explain it anyway. It is, of course, consistent with everything Dianna said, it's just a different way of putting the same physical concepts into words.
Even in non-relativistic mechanics, the energy of an object depends on the reference frame you are considering. Let's say a ball is at rest with respect to me. It's kinetic energy will then vanish. Now consider another observer, moving with respect to me with speed v. From the point of view of this observer, the kinetic energy of the ball will not vanish. Indeed, it'll be mv^2/2, where m is the ball's mass. So, it only makes sense to compare the energy of two different systems or of the same system at different instants of time if we use a single reference frame for both.
When we consider photons in an expanding Universe, something similar happens. The energy of the photon at different times is changing, but not because it's losing energy to something, but rather because we are looking at it from different reference frames. Now you might ask: why don't we fix a single reference frame once and for all and use it to study the motion of the photon at all times? Well, because we can't. This is one of the surprising facts about general relativity, that reference frames cannot, in general, be defined globally. Each reference frame can only be used to describe events in a limited region of spacetime, that is small enough for its curvature to be negligible. So, if we want to study a photon that has been travelling for billions of years, we actually have to choose a different reference frame to look at it at each instant of time. Since the reference frame is changing, the numerical value of the energy is changing. This shouldn't be more mysterious than the example of the ball that I gave first. In particular, I hope it's clear that it makes no sense to speculate about "where the photon energy is going". It isn't going anywhere. If, from the point of view of a single reference frame, the energy of an object diminishes with time, then OK, some other physical entity must be receiving this energy. But this is not what is going on with the photon. The energy changes because the reference frame changes, that's all.