General Relativity gives us a wide range of choices about what space-time coordinates to assign to different events. There's no definite fact about which events are simultaneous, for example, but rather a wide set of choices about how to draw same-time-sheets through the event space. As I understood it, your original question was about what sort of choice people usually make for those sheets when they're describing cosmology.
Let me start with your new questions. You ask "was everything moving at the same speed?" That question is meaningless. Any object with any rest mass can be assigned a velocity of zero, and then the rest of the coordinates stitched together around that. You can end up assigning any velocity up to the speed of light in any direction to any object, depending on your choice of coordinate frames.
The phrase "co-moving" does not describe the behavior of the objects. The overall picture of the Big Bang is that, picking a typical lump of stuff to call "stationary", distant stuff will be moving away from it at a roughly constant speed. On top of that overall picture, there are of course all sorts of local motions as things whizz around each other.
Instead by "co-moving" what we mean is that in assigning a time coordinate to some event we pick the time that a clock which traveled along with the average of the things in the neighborhood of that event would read. As I mentioned, the advantage of this choice is that if we look at all the events at time, say, one year post-Bang, they all look like the stuff there is the same age. For other coordinate choices the things at "one year" look like different ages, depending on their locations.
There's a Wikipedia article that describes these coordinates more completely, but in somewhat technical language. http://en.wikipedia.org/wiki/Comoving_distance
(published on 05/05/12)