At the center of the milky way, there must be some giant mass that keeps the galaxy from just floating away from itself. So, obviously, we feel the immense gravitational pull of whatever this is. But, on a smaller scale we can see this with the earth and the sun. The sun keeps us from spinning into oblivion with its gravitational pull. The question I am asking is, why do we orbit the sun, but not the center of the galaxy, which has a much stronger gravitational pull?
The so called inverse square law makes gravity dominate locally. The Moon orbits Earth, Earth orbits the Sun, and the Sun orbits the Galactic center! Because the distance from Earth to the Moon is 1/500 the distance from Earth to the Sun, it is Earth's gravitational field which dominates the orbit of the Moon.
And it is not a single thing which attracts mass to orbit the galaxy. It is the sum of mass of everything in it. The center is of course the galactic center of mass.
There are a number of misconceptions with this question.
First off, the "gravitational pull" (which I'm interpreting as gravitational force) by the Milky Way on the Earth is seven orders of magnitude smaller than that exerted by the Sun.
Secondly, gravitational force is the wrong metric. The Newtonian gravitational force exerted by the Sun on the Moon is more than twice that exerted by the Earth. Yet all but a tiny, tiny minority of professionals who deal with the solar system will say that the Moon does indeed orbit the Earth.
Thirdly, it's not correct to look at "orbit" as being a mutually exclusive concept. That the Moon does indeed orbit the Earth doesn't mean that it doesn't orbit the Sun. It does. The Moon not only orbits the Earth and Sun, but it also orbits the Milky Way, the Local Group, and the Local Supercluster.
So what is the right way of looking at orbits amongst a hierarchy of masses? One issue that needs to be addressed is what the word "orbit" means. Note that people do write and talk about parabolic and hyperbolic "orbits". I'm assuming that "orbit" in the context of this question means a bounded trajectory, which would rule out calling a hyperbolic or parabolic trajectory an "orbit".
That leads to the first of three characteristics of whether object A can be said to be orbiting object B: Object A can be said to be orbiting object B only if object A is gravitationally bound to object B. The appropriate metric here is energy, at least for cosmologically small distances. (Note: The distance between the Milky Way and the Andromeda Galaxy is very small in a cosmological sense.) From the perspective of a frame centered at object B, the total mechanical energy of object A needs to be negative he mechanical energy to be able to say that object A is (at least temporarily) orbiting object B.
Bounded orbits remain bounded forever in the two body point mass problem. That may not be the case when other larger objects are involved. This leads to a second criterion: Object A can be said to be orbiting object B only if the gravitational influences of larger objects on object A are but small perturbative effects that don't result in instabilities. The appropriate metric here is an energy-based sphere of influence. The two most widely used spheres are the Hill sphere and the Laplace sphere. For example, because the Moon is gravitationally bound to the Earth and because the Moon is well inside the Earth's Hill and Laplace spheres with respect to the Sun, it is appropriate to say that the Moon does indeed orbit the Earth.
What about the Sun and the Milky Way (and even larger objects such as the Local Group)? Can the Moon also be said to orbit those? The answer is yes. One way to look at it is to ask what would happen if the Earth suddenly disappeared. Would the Moon's path around the Sun change much? The answer is no. The Moon would still be orbiting the Sun at about one astronomical unit. The same applies to the Earth with respect to the Milky Way. The Earth's trajectory about the galaxy would not change by much at all if the rest of the solar system magically disappeared. This point of view however leads to some troubling issues. Planets (and stars and galaxies) don't magically disappear. Moreover, if Saturn somehow did magically disappear, its rings and all but its very outermost moons would be on hyperbolic trajectories with respect to the Sun. A way around this magic is to say that if object A orbits object B, and object B orbits object C, then object A also orbits object C. "Orbit" is not a mutually exclusive concept.
The stars do orbit the center every 250 Million years according to a galactic year, That's 20-25 times since the formation of the solar system.
The complex shape of the galaxy, with a disk, bulge, bar and arms adds many non concentric gravity poles to the equation. Easiest to imagine is a wave of movement relative to the disk. It would be nice to see a gravity map of a galaxy in 3D to see that it is not a 1 body at the center kind of gravity map.
Some (many? n%) stars can result from the pressure front of supernovae that happen inside star nurseries, which gives them erratic vectors compared to the disk and the arms. Perhaps if there wasn't the constant energy of supernovae affecting the birth and movement of objects in the galaxy, there would be rings as seen for Saturn.
There's a parallel with mapping rocks in the solar system. If you measure the denser zones to try to figure out where every rock was moving, you'd find your predictions soon become chaos of spirals and local deflection, rather than simple trajectories of a predictable fashion.