How big is the Universe?

It's a bit more complicated than that.The expansion of the universe is not primarily due to movement, but to the expansion of space itself. This expansion is not limited by the speed of light. Further there is nothing to say that at the big bang the size of the universe was limited.

Not quite. Under our current theories every gets closer and closer together as time goes back approaching a theoretical point where everything is infinitely close together. However, the actual science of the big bang begins something like 10^-30 seconds after this point. At that point all the distance we measure now were compressed to a incredible degree: points billions of light years apart now were nanometres apart. But an infinite number of nanometres is still an infinite distance, so it's quite possible for the universe to be infinite at this time.That's as I understand it anyway. Hopefully, CaveDiver will be along in a bit to explain why I've got everything completely wrong.