I am no physicist, but I still feel as though this is an immensely misleading subject.
The idea of an infinite universe conjures up images an infinite number of worlds out there, infinitely many of which must be similar to our own. And somehow, all that infinite matter/energy must have been compressed infinitely densely at the beginning.
But isn't it the case that we have only measured the flatness of space to within about half of one percent. If so, it seems much more likely that the universe is locally very flat but still finite, though very large, due to initial inflation (the event, not the process). So the universe might be 10^100 light years across, but still finite, yet definitely flat to well within half of one percent.
But let's suppose that the universe is infinite (since we have not, and may never rule it out). What does this actually mean? Does it mean an infinite number of galaxies out there somewhere at this moment in time? Somehow I doubt it.
What does it even mean for a galaxy to be out there at some point at this time? Do we mean time relative to the local reference frame of that galaxy (stationary in that frame), say with respect to the big bang or time relative to our own reference frame, which we agree is moving rapidly with respect to the other?
Even distance measurements vary between the two frames of reference.
So my guess is that to even have a concept of what this all means, we need to be expert at understanding space-time manifolds, which we are famously hopeless at.
Another obvious point: it isn't possible for an observer to travel off in one direction and return to the same point, even if the universe is curved (as I suspect it is). The reason is simply that you couldn't make that journey. There are points in the universe which lie out of causal contact with us, which means that there is no means of getting there, even theoretically. We can't even observe those points, let alone go there and see them. So we can never make the full journey in one direction and return from the other. Doing so would imply travelling beyond our cosmic horizon.