Re: If Space turns out to be discrete, what lies in between

Well, the basic idea is that there exists something called the Planck length; this length, about 10^(-35) meters, which is calculated from a combination of fundamental constants (the speed of light, Planck's constant, and the gravitational constant), is the length at which quantum gravitational effects become important. Another way to put it is that you cannot measure lengths any shorter than this, nor can any particle be confined to a space any smaller than this. (There is also a Planck time, calculated from a different combination of the same constants; the same arguments apply so I won't discuss it separately).

At this point we are dealing with the quantum mechanical world, which is quite different from our everyday world! The question of how you get from one bit of space to another is similar to the question of how an electron gets from one energy level to another within an atom - it just does. You can never measure it as being anywhere except at a particular discrete energy, so it makes no physical sense to talk about it "passing through" other energies on its way. Similarly, if you can never measure a length shorter than the Planck length (due to how space works, not due to any limitations on your ability to measure things), shorter lengths have no physical meaning.

There seems to be considerable agreement that the Planck length is where one needs to describe spacetime using quantum gravity (a theory which at the moment does not exist in any useful form); whether or not this implies an actual quantization of spacetime is not quite so universally accepted.

Here are some references on the Planck length (from simpler to more complicated):

What is Planck length? What is Planck time?
http://www.physlink.com/Education/AskExperts/ae281.cfm

Can things be infinitely small?
http://itss.raytheon.com/cafe/qadir/q432.html

How Is the Universe Built? Grain by Grain
http://faculty.washington.edu/smcohen/320/GrainySpace.html

The Planck Length
http://math.ucr.edu/home/baez/planck/node2.html

And here is an artist's rendering of what spacetime might "look" like at the Planck length (to be taken with many grains of salt - you cannot actually "see" this, of course). It's about 3/4 of the way down the page:

Artistic conception of spacetime foam
http://www.physics.wustl.edu/~visser/pictures.html

Finally, here are some tutorials on general quantum mechanics (to address the issue of how to get from here to there without passing through the points in between):

What is a Wave Function? What is an Orbital? An Introduction to Quantum Mechanics
http://www.chemistry.ohio-state.edu/betha/qm/

Quantum Mechanics Introductory Tutorial (requires Shockwave)
http://www.science-spirit.org/quantumtutorial/quantum.html

Rudiments of Quantum Theory
http://www.chembio.uoguelph.ca/educmat/chm386/rudiment/rudiment.htm