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As the universe expands, we notice almost all matter moving away from us thanks to a spectral shift to red. Yet my question is that if every gallaxy is moving away from one another, why are we doomed to eventually crash into our closest galactic neighbor, Andromeda?

So you're saying that our galaxies are moving faster than the universe is expanding? that doesnt seem very plausable.

My guess was that our local galaxy cluster was free standing, mostly held together with gravity. and all other clusters were moving away from us.

One more questions. If space itself is expanding, does that mean we are expanding as well? We do dwell in space. Who says expansion is only targets big things and the space between

I asked my Physics teacher the same thing. He said the universe is expanding like raisins in a loaf of bread. The smaller things don't expand; they just move apart from one another.

The space between your atoms is expanding, but the chemical bonds between your atoms are pulling the atoms together, like if a pair of the raisins were tethered to each other.

How space pulls things apart from each other depends on how far apart they are. If two things are close together, space pulls them apart slowly; things far apart get pulled apart very quickly. So your atoms are only being pulled apart by the spatial expansion very slowly (about an atom's width every 10 billion years), and the chemical bonds are strong enough to overcome this easily.
Even galaxies within our own galactic cluster are very "close together" from the point of view of spatial expansion. Andromeda is about 2.5 million light years away from us. According to Hubble's law, the space between us and Andromeda is expanding at a rate of about 10 kilometers per second, which isn't very fast from an astronomical perspective. In comparison, Earth moves at about 30 km/s around the sun.

Unless the two objects are separated by billions of light years, it's not that difficult to overcome the expansion of space.

One way to think about the link between distance and expansion rate is to consider three raisins in a line, A, B, and C, so that B is perfectly in the middle between A and C. At any given instance, the distance between A and B gets bigger at one unit per second, and the distance between B and C gets bigger at one unit per second, and so the distance between A and C must be getting bigger at a total rate of two units per second. Hence the distance between A and C grows at a faster rate because it has deal with the expansion between A and B and the expansion between B and C. So the rate at which the distance between two objects grows depends on how far apart the objects are to begin with.

On a final note: it's probably not for the best to think of things getting smaller. There are certain physical constants that involve lengths that don't expand with space, and we don't want to have to scale these values to deal with "shrinkage" if we don't have to.

Actually, if anything it's speeding up; that is to say that the ratio of velocity to distance is getting bigger.

The current model has that expansion was slow right after the Big Bang, then there was a short period where the expansion was really, really fast (this is called "inflation"), and then the expansion slowed down again, but has been getting faster ever since. So eventually, the distances between galaxies in our cluster will be enough so that the distance between them grows faster than the speed of light. If the acceleration of the expansion keeps up, eventually even a measly 2.5 million light years will be enough so that two galaxies separated by the distance will retreat from each other faster than light. At this point not even light from these galaxies will be able to reach us; it will appear as if our galaxy is the only one to exist.

In fact, if the acceleration keeps up for long enough, eventually human-sized distances, in kilometers and meters, will be enough so that other people are dragged away from you faster than light. And then your own body parts will retreat faster than light, faster than your ligaments and tendons can repair, and then your cells will be pulled apart by the expansion of space. Biology will cease to mean anything.
And then your molecules will get pulled to pieces as atoms begin separating faster than the speed of light, faster than chemistry can act.
Then the distance between electrons and the nucleus will become the distance at which the electrons and the nucleus separate faster than light, at which point the electrons will become permanently separated from the nucleus; and then the nucleus will be torn apart, the pieces unable to attract each other faster than the expansion of space drags them away. Then individual protons and neutrons will fall apart.
At this point, no particle is able to interact with any other particle, because interactions can't go faster than light. Physics doesn't mean anything at this point.
This is called the Big Rip, a very unpleasant hypothesis about the end of the universe. It also currently appears very, very likely.