How fast does the sun move in space?

I know we live in an aparently expanding universe, but is there any way of
working out how fast the sun is moving relative to space itself, ie not just
another object in space? Why would it matter? well lets imagine that in
fact it is not the whole Universe that is expanding but that the big bang
was just one particular piece of matter that exploded and we are part of the
debris moving outwards. But say there were other objects already in space
that were not moving., or perhaps were part of another big bang somewhere
else and are moving in a completely different direction. At some stage
could we not hit them at some enormous speed?
terry