Why study metric spaces? Most universities have a 3rd year undergraduate analysis course in which metric spaces are studied in depth (compactness, completeness, connectedness, etc...). However, in practice it seems that most of these metric spaces are normed vector spaces. Why not just cover normed vector spaces instead of metric spaces? 
Even if we lose some generality, normed vector spaces feel more natural and interesting, in my opinion, at least. 
 A: Metric spaces are more general than normed spaces, because they need not be vector spaces. They are easier than general topological spaces, but introduce all of the relevant concepts.
A: You write in a comment that metric spaces "don't come up very much in practice".
This is not true (although may reflect the mathematics you have seen so far).
Metric spaces (and, more generally, topological spaces) occur all over the place.  I am a working number theorist, and I use the concepts of topology (in all kinds of contexts, sometimes in the context of vector spaces or rings or groups, sometimes in very non-linear contexts) all the time.  Geometers and topologists use them still more frequently (perhaps unsurprisingly).
The language and basic results of topology (open and closed sets, continuity, connectedness, compactness) are some of the most flexible and useful concepts in 
mathematics!
Added: In my answer I've tended to conflate metric spaces and general topological spaces, but perhaps in your question you want to distinguish them.
(Maybe you are wondering where particular metrics arise that are not induced by
normed spaces.)
To the extent that geometry is about studying lengths, angles, and related concepts such as curvature, it is very much a subject that revolves around metric spaces, and in modern geometry, geometric topology, geometric group theory, and related topics, many techniques use metrics as the basic strucure.
E.g. Gromov--Hausdorff limits.
E.g. Metric space approach to concepts such as curvature, leading e.g. to
CAT-0 spaces.
E.g. You might be tempted to think as Riemannian geometry as being a more
analytic than combinatorial/metric-space based subject, because of the role
of differential topology in the foundations.  But metric space notions
(such as the two previous examples) are fundamental in modern aspects
of the theory, such as rigidity.
A: Metric spaces are far more general than normed spaces. The metric structure in a normed space is very special and possesses many properties that general metric spaces do not necessarily have. 
Metric spaces are also a kind of a bridge between real analysis and general topology. With every metric space there is associated a topology that precisely captures the notion of continuity for the given metric. That means that many topological properties can be understood in the context of metric spaces, encompassing many many examples of varying degrees of complexity, while still having a notion of distance. The notion of distance is typically seen as much less abstract than that of a topology, with many of the proofs just being a re-formulation of the standard proofs from real analysis. In fact, somewhat less known is that a naturally occurring generalization of metric spaces (i.e., allowing the metric to attain values not just in the range $[0,\infty]$ but rather in what is known as a value quantale) is as general as topological spaces. That is, every topological space arises as the associated topology for some metric structure in that sense. 
Also, many universities are somewhat reluctant to introduce abstract notions early on and so even though the limits in real analysis can be taught just as special cases of metric analysis, and thus introduce metric spaces in the first year, universities often choose to postpone metric spaces. However, this is not always the case as I had the opportunity to teach metric spaces in a first year course (at Utrecht University). 
So, this is largely a question of tradition and certainly varies between universities. As is commonly the case, university curricula tend to follow historical developments and tend to adapt and change rather slowly. Metric spaces were introduced in 1906 by Frechet and thus are quite newer than the traditional primary objects of study such as $\mathbb R$ and various function spaces. This may explain why the latter are often taught first. 
A: Another perspective might be useful for others looking at this question. Studying subsets of normed spaces is essentially the same as studying metric spaces. Given a metric space $(M,d)$, we can define a vector space $L^\infty(M)$ of bounded functions $f: M \to \mathbf{R}$ with domain $M$. Given a point $x \in M$, consider the function $f_x$ given by $f_x(y) = d(x,y)$. Then for any pair $x_1, x_2 \in X$, $f_{x_1} - f_{x_2} \in L^\infty(M)$, and moreover, $d(x_1,x_2) = \| f_{x_1} - f_{x_2} \|_{L^\infty(M)}$. Thus if we fix $a \in M$, and consider the map $F: M \to L^\infty(M)$ given by $F(x) = f_x - f_a$, then this identified $M$ with the subset $F(M)$ of $L^\infty(M)$. So it is natural that most metric spaces you can think of can be naturally identified as subsets of norm spaces, because all metric spaces can be made a subset of a norm spaces, albeit maybe not all `naturally'.
