Should quantum mechanics be learned first before relativity?

They are independent enough to be taken in on in either order, but the more natural progression is to do special relativity first. Two reasons, one practical and one a matter of taste:
1) QM requires much more math than SR, as well as a decent understanding of classical wave mechanics (that's a one-semester topic in it's own right). So you can start in on SR while you're still picking up the math skills for QM.
2) For anyone who is not planning to be a practicing physicist, SR will be more valuable. It's a lot easier to make the case that SR should be part of every well-rounded person's education than QM.

 

I put SR into the list of things that make us better thinkers, even if we don't use them directly: things such as learning a foreign language, being familiar with the literature and history of your culture, playing a musical instrument, basic economics, theory of evolution, history of philosophy, .... You don't need any of these, but almost no matter what you're going to do in life, you'll do it better and get more out of it with these experiences.

The great value of studying SR, for anyone who is not going to be using physics in their day job for the rest of their life (and that's nearly everybody), is the experience of rigorously and logically analyzing and then moving beyond our intuitions about how the world works... and it's accessible to anyone with even high-school algebra!

I agree that QM has more practical applications; it is essential to understanding why atomic bonds, semiconductors, chemistry all work the way the way they do. However, if we look at what most practitioners in these fields are doing... They don't need to and aren't bothering to really seriously understand QM, they only need to accept the results and apply them.

I also think that there is much to be said for studying QM in the context of philosophy, working through the problems of interpretations and observation and decoherence and Bell's theorem... Even if you're going to be a lawyer, you'll be a better lawyer for having done that study. But you don't have to invest the skull sweat involved in understanding the hydrogen atom, let alone some more complex system, to do that.

 

This is indeed true, and the effect is that a QM course is even less valuable to any non-specialist - if you aren't going to be doing these calculation later in life, what are you getting out of the time spent learning how to do them?

By contrast, just about anyone who can hack the (not nearly as demanding) math of SR will benefit from making the the effort to learn SR. It improves general thinking skills in the same way that learning another language or studying literature does.