Designating a coordinate system with multiple objects So I am slightly new to physics, but am thoroughly enjoying the contextual thinking changes that physics brings about. My question is regarding coordinate system designation, on a group of individual masses that move as one unit across differing inclines.  What is the best way to assign a coordinate system and does the system stay uniform throughout the entire free body diagram or does it shift with a shift in directional forces?  If someone could explain this I would greatly appreciate it.
 A: Any inertial coordinate system is valid here. One might choose one orientation over another if it simplifies the problem (reduces trigonometry needed). In your case, it isn't clear what you are looking at, so one cannot answer as to which coordinate orientation is preferred or not.
A: 
(Image source: https://mech.subwiki.org/wiki/File:Pulleysystemondoubleinclinedplane.png.)
Something like this, but with $m_1$ on a flat surface?
I recommend using separate coordinate systems for the two objects because you will want to draw separate free-body diagrams for each. For $m_1$, arrange the $x_1, y_1$ system with the $x_1$-axis parallel to the surface on which $m_1$ slides. Then do the same for the $x_2, y_2$ system for $m_2$, so that the $x_2$-axis is parallel to the surface on which $m_2$ slides.
I think the uniformity you're thinking about is this: each object experiences the same magnitude of tension in the connecting string, and each object experiences the same acceleration if the string doesn't expand or break. Also, the positive and negative directions of the coordinate systems should be correlated to make things easier-- so that the if the tension experienced by $m_1$ is in the positive $x$-direction of its coordinate system, the tension experienced by $m_2$ is in the negative $x$-direction of its coordinate system. (It isn't necessary to do this to get the correct answer, but it definitely makes things easier.)
It is possible to conceptualize problems like this in such a way that the string and pulley are ignored, and $m_1$ and $m_2$ are combined into a composite object of mass $\left(m_1 + m_2\right)$, which experiences only the external forces (not the tension) acting on $m_1$ and those acting on $m_2$. In this context, there will be only one coordinate system, and correlating the signs between the original, separate systems will help a lot in setting this up. The internal force, the tension in the string, can be ignored in the same way that we are ignoring the internal forces between the molecules of each object. In my experience, this is a good exercise for physics students studying problems like this.
Another good exercise it to not neglect the pulley, and assume that it rotates as the blocks slide. Some of the initial gravitational potential energy of the system must go into the rotational kinetic energy of the pulley, which means the acceleration of the blocks will be less than that calculated when the pulley is ignored. Just assign a mass $m_3$ and radius $r_3$ to the pulley, and perhaps assume that $m_3$ < $\left(m_1 + m_2\right)$.
Both of these extensions are best reached by starting with separate free-body diagrams as I described above. Solve that problem first.
Please feel free to post any additional questions.
Hope this helps!
