Should I use Mathematica to understand circuit behavior?
Understanding where the equations come from is definitely important, but having an intuitive sense of what a circuit is doing when you look at it is more important.  That way, you can compare what the equations indicate with your intuition about the circuit and see if they disagree.  If they do, then either your intuition is not entirely correct or you made a math error in deriving the equations.  

In this case, the output is rather easy to explain.  An ideal diode only allows current to flow through in one direction.  When the diode allows current through, it dissipates some voltage Vd, which is usually around 0.6 or 0.7 volts for a silicon diode.  This is called the 'forward voltage' or 'diode drop' and it is more or less independent of current.  Actually, it is exponentially dependent on current, so once you get enough current flowing it really doesn't change all that much and so can be approximated as constant.  Note that this is an approximation and will not be true in all cases.  When the voltage at the input of this circuit is less than Vd, no current flows and the output is 0.  Once the input voltage exceeds Vd, then the output will be Vd less than the input as the diode absorbs Vd of the input voltage.  

I would not be too concerned with using something like Mathematica to generate an answer like that.  For very simple circuits, it can be useful.  But for more complex circuits, it can be nearly impossible to derive equations that exactly describe the operation of the circuit.  For complex circuits, the best you can do is either make approximate calculations to figure out more or less what the circuit does, or use a circuit simulator like SPICE to get a numeric approximation for a particular stimulus.  There are a lot of approximations that you can make to simplify circuits and determine what they do with some very simple calculations, but it takes a bit of experience to figure out when various approximations are applicable.  Using Mathematica or various simulations can help validate your approximations.