|MadSci Network: Astronomy|
First things first. The Big Bang happened when all of the matter in the Universe was concentrated into a single point. There was no Milky Way Galaxy and there were no points of reference from which to measure a speed. The measurement of speed requires a frame of reference. The Universe was initially a very hot and dense plasma. It did not cool enough for matter to begin to collapse under gravity for millions of years. The first protogalaxies did not begin to form until about a billion years after the Big Bang. So only at a billion years old can we even start to pick out a location in the Universe where the Milky Way is forming. But, if you want to ask how far the Milky Way has moved since it first formed (~10 billion years ago) again the problem of reference points is an issue.
For the moment let us forget about gravity and assume that the Milky Way is just cruising along, taking part in the smooth expansion of the Universe and not being slowed down or sped up by large clumps of nearby matter. In this situation the Milky Way never leaves the location from which it formed. In the Big Bang model the Universe did not explode into anything ('Big Bang' is actually a misnomer) but rather space itself is expanding. We observe other galaxies to be speeding away from us, the farther away they are the faster they seem to be going. Observers on those other galaxies observe the exact same thing, they see us moving away from them with some speed proportional to how far away we are from them.
Now, you originally asked how far we had moved since the Big Bang. In oder to figure this out we would only need to know how fast the galaxy has been traveling and for how long it has been going at that speed (assuming that it has not speed up or slowed down). But what speed are we going to use? Originally all of the matter in the Universe was at the same point. Pick a galaxy from the Virgo cluster, it is currently about 17 Megaparsecs away. It was once much closer, in fact at the Big Bang we were both at the same location. So in about 12 billion years we have moved 17 Megaparsecs away from each other. But is that how far we have moved since the Big Bang? No. Pick another galaxy, say a quasar currently 1 billion parsecs away. It was once much closer. In fact at the Big Bang we were at the same location. So in about 12 billion years we have moved 1 billion parsecs away from each other. So, is that how far we have moved since the Big Bang? Again, no.
So there really is no simple answer to your question. Gravity from nearby concentrations of matter does affect the motion of the Milky Way and makes it deviate slightly from the smooth expansion of the Universe. But figuring out how much it has deviated from its birth place because of that is very complicated and not well known by anyone at this time.
So when you ask how fast was the Milky Way initially moving, I have to respond by asking "relative to what?" Likewise when you ask how far the Milky Way Galaxy has moved since the Big Bang, I must again ask, "relative to what?" From the examples above you can see that the answer depends upon what your point of reference is.
If I have missed the point of your question altogether please let me know, but I suspect that you have an incorrect picture of the Big Bang in your mind as having been an explosion of matter into an already preexisting space. This was not the case. Space began expanding at the Big Bang, and it is not expanding into anything.
Try the links in the MadSci Library for more information on Astronomy.