Where is the center of the universe? Do we know the location of the big bang and would that intuitively be the same place?

As my mother reminded me repeatedly as a child... It ain't me.

And according to some theories (many) there actually is none. The universe IS expanding, and that expansion is the same everywhere. A quote from the Physics FAQ by Scott Chase, Michael Weiss, Philip Gibbs, Chris Hillman, and Nathan Urban might help:

There is no centre of the universe! According to the standard theories of cosmology, the universe started with a "Big Bang" about 14 thousand million years ago and has been expanding ever since. Yet there is no centre to the expansion; it is the same everywhere. The Big Bang should not be visualised as an ordinary explosion. The universe is not expanding out from a centre into space; rather, the whole universe is expanding and it is doing so equally at all places, as far as we can tell.

In 1929 Edwin Hubble announced that he had measured the speed of galaxies at different distances from us, and had discovered that the farther they were, the faster they were receding. This might suggest that we are at the centre of the expanding universe, but in fact if the universe is expanding uniformly according to Hubble's law, then it will appear to do so from any vantage point.

For more info check out http://www.desy.de/user/projects/Physics/index.html