If you somehow got past all of the matter in the universe, you wouldn't ever hit a wall. The universe is like an infinite room with a bunch of matter in a finite (though unfathomably large, by our standards) part of the room. At least, that's from what we can observe. There is always the possibility that there are other universes beyond our own that we haven't seen because the spacetime between our universes is expanding faster than the speed of light (spacetime's expansion is not limited by the speed of light; only objects moving within spacetime are), and if that's the case then we'll never be able to see them, let alone get to them.
And the Big Bang theory doesn't predict that the universe is constantly expanding then collapsing. Or rather, there are three possibilities presented by the Big Bang: that the universe is either Open, Closed, or Flat.
In an Open universe, everything will continue expanding forever, only slowed slightly by gravity.
In a Closed universe, gravity would eventually overtake the rate of expansion. It would, at some point, stop expanding and start contracting until eventually it all collapsed back into itself.
In a Flat universe, the rate of expansion would slowly approach a rate of zero as an asymptote. It would neither collapse nor infinitely expand. After an infinite amount of time, it would reach a finite volume.
Through the Cosmic Microwave Background radiation and mathematical modeling of what the CMBR should look like in an open, closed, or flat universe, we now know that the universe is Open; that it will expand infinitely.