Is the North star going to stay in the North sky? Considering how the our solar system, and how every other star in the galaxy is traveling through the cosmos, it seems like it's a miracle that the stars and constellations in the sky stay as constant as they are.  



How is it that the North Star seems to be fixed in space? Is it following our star through space? are we following it? Is it actually moving? Are we getting further away from it? Closer to it? What is our relationship to Polaris in the Galaxy?
The relative direction to an object depends mainly on two things:


The location of the observer and the object
The coordinate system you use to determine direction.


So the first part of that concerns where we are compared with Polaris (and by extension of your question, where the other stars are as well).  Yes the stars and other components of the galaxy all have certain motions.  But this motion is not large compared to the distances between stars.  

Most of the stars we can see in the sky are quite nearby, and are orbiting the galaxy at similar speeds.  The speed of the solar system around the galaxy is around $230 km/s$.  The relative speed between our sun and nearby stars is much less than that.  If we assume that most stars are farther than $5ly$ from us, then we can calculate how long it might take to make a $1\%$ change in distance.
$$t = \frac{d}{v}$$
$$t = \frac{4.7x10^{23}m}{2.3x10^5m/s}$$
$$t = 2.06x10^{11}s$$

That works out to over six thousand years.  There's a lot of caveats with that number, but it gives you a good idea that for the most part it takes too long for stars to move relative to each other for you to notice during your lifetime without very careful observation.

The second point is that things can move if we change what we mean by "north" or any other direction.  North, both on earth and in astronomy, is defined by the rotational axis of the earth.  But the axis changes its direction with respect to other stars over time.  Besides some smaller scale wobbles, the largest change to the direction of the axis is due to precession and has a period of about 26,000 years.  This change will take "north" from a direction that passes relatively near Polaris in a large circle around the sky.  With such a long period, again this isn't something to be seen by individuals without detailed measurements.  



So the answer is that people alive today will be able to always think of Polaris as the pole star, but this will be less true for future generations as changes are easily visible on the scale of thousands of years.