The first thing you need to understand is that the big bang did not occur at a particular point in space. Expansion occurred everywhere at the same time. At about t=0, the point in time that we say the big bang occurred, the observable universe was very small in size. However, there was still much more of the universe than that, possibly an infinite amount of space, where matter was highly compressed and very, very hot. Over time the matter and radiation in the entire universe, both the observable part and the regions beyond, expanded and became cooler and less dense over time.
The light that we see now did not come from our own section of the universe, but was emitted a very long time ago from regions of space far away from ours. We actually can't see back beyond a point in time about 186,000 years after the big bang. Prior to this point in time, the universe was too hot and dense for light to freely propagate. It was continuously emitted and absorbed, never getting anywhere. Then, about 186,000 years after the big bang, an event called "recombination" occurred where most of the hot plasma finally cooled off enough to turn into neutral gas, which is transparent to light. Once this occurred the very first light was emitted that we can still see today. This light has been redshifted over time and is now in the microwave range. It is known as the Cosmic Microwave Background.
Note that the nature of expansion means that everything expands away from everything else not bound to it. From our point of view, everything in the universe is expanding away from us. The light we see today from old galaxies was emitted long ago when those galaxies were much closer to us.
Does that help?