MarkStephen, if you try to visualize this, you will go nuts. Some modern cosmologies go up to 10 dimensions, and we can only try to visualize a 4th by analogy with two dimensions curved into a third!
You can only get an idea of how it hangs together by going deeply into the maths.
I cannot imagine infinite space, but I cannot imagine an end to space, either. A finite universe in an infinite space means that the emptiness just goes on and on and on and... Horrors! But a curved space does not help - does the fourth dimension go on for ever? String theory tries to fold up dimensions into microscopic strings which may be light years long! You can read the books, but don't hope to understand it.
I have a problem with a finite universe expanding into empty space, because then it would have started off as a black hole, and expansion would have been impossible. But the general (?) view is that space is expanding, taking the universe along with it, and this is supposed to get around the problem somehow.
Don't worry about infinite spaces expanding. You can double the size of infinity, and you still have infinity.
By the way, time is usually regarded as the fourth dimension, so space would have to be curved in the fifth dimension.
Feel confused? Join the club.
Mike.
™ Version 4.2