So I was reading this answer about how galaxies are the fastest moving objects in the universe because space is expanding faster than the speed of light. This got me wondering, would it be possible to be 'still' inside of space so that galaxies are moving away from you and others towards you? If so, wouldn't this allow us to travel from place to place in the universe much quicker? I know that this wouldn't be something that's even close to possible today, but I was just curious if such a thing is possible. Thank you.
The so called expansion of the universe is not as trivial as most people think. What is happening, in fact, is that the distance between two points in space (note that I'm not talking about objects with velocities, but just coordinates in space) increases with time in a manner proportional to a given factor (in this case, the Hubble constant - which is actually not a constant in time, but let's ignore that for now).
This means that galaxies are not exactly moving away from a specific point with a particular velocity (i.e., a particular rest frame), but that the space between them is expanding. It is somewhat hard to wrap your head around the difference between these two situations, but this has to do with the geometry of space, and not with what's inside it.
It is widely accepted that the expansion of the universe is homogeneous and isotropic (the Cosmological Principle), which means that there is no "special" position (rest frame) on the universe, and whatever your velocity and your location, the universe will seem to expand the same way.
The best way to imagine this is to reduce to 2 dimensions expanding in a third dimension. For example, suppose that you live in a galaxy embedded on the surface (a 2D universe) of a balloon (completely unaware of the third dimension), with other galaxies distributed homogeneously. If, for some reason, the balloon is expanding isotropically and, every point of the surface will be moving away from every other point, so will every galaxy from each other. Think about this: is it possible to find a special, "still", point (without resorting to the elusive third dimension) where you will see galaxies moving from you in one side and galaxies moving towards you on the other side?
Our universe works similarly, but we're embedded on a three-dimensional space instead of a two-dimensional one.
There is no 'still' that is not relative to some other object. So yes, you can be still with respect to one object, but you'll be moving with respect to every other object.
As for using it as a form of travel, letting your destination come to you, well, there's no difference between that and simply moving toward your destination other than the way you phrase it.
We are moving "... relative to the comoving cosmic rest frame ... at some 371 km/s towards the constellation Leo". That's slow in comparison to the speed of light. ...And it would be possible to be "still" relative to the "comoving cosmic rest frame": Just travel with 371 km/s to the opposite direction relative to the sun.
Distant galaxies are moving fast relative to us due the expansion of space. They are moving slow relative to the cosmic microwave background seen from their location. Slowing down relative to one of those distant galaxies would mean to accelerate to a high velocity relative to our local environment, and relative to the cosmic microwave background.
The effect would be as you described: Some galaxies would approach, some would move away, according to the direction we would travel.
The best definition of "still in space" you can have is that you make a weighted average of galaxy velocities around you and their mean movement is zero. This means all of them recedes from you at the same speed when at the same distance, so you are "still" regading what we can call "the cosmic expansion frame".
Note that you must discard nearby galaxies, where their proper motion is big. And note also that you will not be still in reference to some other viewer that perform the same experiment. In fact, you both will be receding from the other at the cosmic expansion rate.