$r^2 = 35\%$, $r = 0.59$ How does a (pro) statistician formally interpret this correlation? Strong? Weak? For fun and amusement, you wonder how well student grades in Course 101 can predict scores in Course 102.  So, you do a regression on grades from both classes.  
If you got an r = .59 with $r^2$ = 35%, would you consider this strong or weak?  Can you say anything more than concluding, "There’s obviously an association, but there are other variables in play, as well."   
Bottom line, just how noteworthy is an r=.59 considered?  If 32% of the 102 grade is “explained by” the 101 grade, that seems like a non-trivial factor.  
TL;DR:  How does a professional statistician formally interpret r=.59 with $r^2$ =35% ?
 A: If you were to blindly apply Cohen's criteria for describing the strength of a correlation, an r = .59 would be described as a "large" effect to have been observed.  
However, you should note that this "large" effect might not actually be statistically significant or even that meaningful.  For the former argument, if you look at the critical values for a correlation coefficient, you need more than 9 degrees of freedom to have this be "statistically significant" at $\alpha < .05$.  For the later, you should look at the scatterplot and really look at how the scores are falling - like the above commenters mentioned, the relationship might not be linear or that useful.
Of course, if the linear relationship does seem tenable ande you have the correlation, you can also generate the linear regression equation to predict someones 102 grade based upon their 101 grade.  But, like Nestor said, this does not mean their 101 grade is causing their 102 grade.
A: What is considered strong and weak tends to differ considerably between disciplines, and for good reasons. It helps to have a correlation in mind from your field that is considered very strong. In my case that would be the correlation of the years of education of spouses (about .60). This gives a bit more substance to a statement that a correlation is "strong": the correlation between grades of two related classes is about as strong as the correlation between years of education of two spouses.
A: Without actually seeing the plot, a 'pro' statistician wouldn't bother interpreting it as a 'strong' or 'weak' correlation because, as you word it, it seems you are implying causality. Remember that dependence sometimes implies (linear, in the case you are referring to the $r^2$ given by a linear fit) correlation, but not the other way around: correlation does not implies causality.
