The FBD's are easy. I am assuming you need to draw the FBD's based on the initial displacements, otherwise you have a total of 8 possible sets of diagrams to deal with. (Two for each spring.)
So for m1 we have a force from spring 1 downward, a force from spring 3 upward, and a weight downward, etc.
Hooke's law states
and positive is always taken in the direction of stretching the spring. So if we displace m1 upward a distance , then the change in force on m1 from equilibrium will be
As far as solving this system is concerned, the simplest method is probably using the Lagrangian method. In the event you aren't up to using that, we need to use Newton's 2nd Law, as usual. Write Newton's 2nd for each mass. So
etc. You will wind up with a system of three linear differential equations in three unknowns (d1, d2, and d3.) This will be "nontrivial" to solve.