Limitations on how far one can travel in the universe

Your question can be translated into "if right now we would send a powerful omnidirectional light pulse from earth into space, would there be galaxies that never see this light pulse?"

The answer is "yes". Due to the accelerated expansion of the universe, as described by the lambda-CDM model, only galaxies currently less than about 16 billion light years (the difference between the cosmological event horizon and the current distance to the particle horizon) away from us will at some time observe the light pulse.

A nice visual representation of this can be found in figure 1 of this publication.

The short answer is yes, there are often parts of the universe an observer can never reach due to expansion. Here are some details:

You're absolutely right that curved spacetime is involved in this question. First let me state an obvious point. Even in flat spacetime, there are always events a given observer cannot reach. Namely, if a bomb sits 1 meter away from you and it is set to explode in $10^{-15}$ seconds (in your initial reference frame), you can't reach it because the speed of light is too slow. The event is said to be outside of your causal future.

However, you can still reach the location where the bomb was sitting before it exploded. It can be done in arbitrarily little time on your clock by moving arbitrarily close to $c$. That is (roughly) the sense in which special relativity allows you to "reach any place in the universe" in arbitrarily small proper time.

In curved spacetime (which describes the expanding universe) a new problem arises. Consider the bomb example discussed above. We'd like to know if you can reach "the location where the bomb sat" in a small amount of proper time. However, this "location" is not a well-defined concept in GR. In completely general spacetimes, the best you can say is that you can't reach events that are outside of your causal future (like the unexploded bomb).

The good news is that in the special case of cosmology there is a notion of the location where the bomb sat. The idea is to imagine that the universe is full of galaxies and to say that two events occur at the same (comoving) location if they are close to the same galaxy. The fact is that in many cosmological solutions, there will be galaxies that follow a path through spacetime that is outside of your causal future. You can never reach them (you may be able to see them though).

An example: our universe is becoming closer and closer to an exponentially expanding universe. In that case, there is a certain distance (16 Glyr for us) such that any observer farther than that away can never reach you. You can think of this as being because the expansion is too fast for the observer to get back to earth.