For example, I find the first group isomorphism theorem to be vastly more opaque when presented in terms of commutative diagrams and I've had similar experiences with other elementary results being expressed in terms of exact sequences. What are the benefits that I am not seeing?

Holy cow, go beyond the first homomorphism theorem! For example, if you have a long exact sequence of vector spaces and linear maps
$$
0 \rightarrow V_1 \rightarrow V_2 \rightarrow \cdots \rightarrow V_n \rightarrow 0
$$
then exactness implies that the alternating sum of the dimensions is 0.
This generalizes the "ranknullity theorem" that $\dim(V/W) = \dim V  \dim W$, which is the special case of $0 \rightarrow W \rightarrow V \rightarrow V/W \rightarrow 0$. The purpose of this general machinery is not the small cases like the first homomorphism theorem. Exact sequences and commutative diagrams are the only way to think about or formulate large chunks of modern mathematics. For instance, you need commutative diagrams to make sense of universal mapping properties (which is the way many concepts are defined or at least most clearly understood) and to understand the opening scene in the movie "It's My Turn". Here is a nice exercise. When $a$ and $b$ are relatively prime, $\varphi(ab) = \varphi(a)\varphi(b)$, where $\varphi(n)$ is Euler's $\varphi$function from number theory. Question: Is there a formula for $\varphi(ab)$ in terms of $\varphi(a)$ and $\varphi(b)$ when $(a,b) > 1$? Yes: $$ \varphi(ab) = \varphi(a)\varphi(b)\frac{(a,b)}{\varphi((a,b))}. $$ You could prove that by the formula for $\varphi(n)$ in terms of prime factorizations, but it wouldn't really explain what is going on because it doesn't provide any meaning to the formula. That's kind of like the proofs by induction which don't really give any insight into what is going on. But it turns out there is a nice 4term short exact sequence of abelian groups (involving units groups mod $a$, mod $b$, and mod $ab$) such that, when you apply the above "alternating product is 1" result, the general $\varphi$formula above falls right out. Searching for an explanation of that formula in terms of exact sequences forces you to try to really figure out conceptually what is going on in the formula. 


If you are asking why very elementary results like the first isomorphism theorem are phrased in the language of exact sequences/commutative diagrams (rather than why this language is used at all), then there are (at least) two answers: (1) for those who are used to using this language, they frequently think about even those elementary results in terms of it, and so it is natural to write them in that language; (2) we want to train students to learn this language, and have to start somewhere, so we begin by taking elementary results that can be understood in another way, such as the first isomorphism theorem, and then rewrite them in this language for pedagogical purposes. If you are asking why people use this language at all (which is to say, why are there many people to whom (1) above applies, and why do we want to engage in the educational practice labelled (2) above), then Keith Conrad gives a pretty good answer. At a slightly broader level of generality, one might cite the old saying "a picture is worth a thousand words", and note that a wellchosen diagram or exact sequence can convey a lot of mathematical information in a succinct and intuitive way (the intuition coming once you have some familiarity with this way of thinking). We have a lot of mathematics to remember, and are always looking for ways to compress our descriptions of things without losing information or becoming unclear. Wellchosen definitions and terminology are one way this is achieved; welldrawn diagrams and exact sequences are another. Finally, one could note that contemplating diagrams appeals (however slightly) to geometric modes of reasoning. Typically, any method which allows one to import some kind of geometric reasoning into algebra is welcome, since it brings less typically algebraic ways of thinking to bear on algebraic problems. 


Putting a proof in terms of commutating diagrams allow you to repeat the same proof in different categories. Maybe you proved something about covering spaces. Reverse the arrows and you can get something about field extensions. Commutating diagrams allow you to separate which part of the proof is purely categorytheoretic and which part of the proof concernings the particular objects and morphisms in the category in question. 


The main advantage that I see comes from generality: the results you're referring to hold for way more types of objects than just abelian groups of modules (for example, chain complexes or sheaves) and the definitions that refer to elements don't make any sense in those contexts. For example, there's a "first isomorphism theorem" for sheaves, but it's more difficult to express it in terms of elements because the definitions of "surjective" and "image" in that category are a little funny. That said, I had a very similar experience when I first started interacting with this sort of language, and a lot of the time whenever I see a statement that involves a diagram, I pretend that everything in the diagram is a module and think about what it means for elements. Eventually you pick up heuristics for all the concepts involved, and you can learn to switch back and forth between the two descriptions. For example, a short exact sequence $0\to A\to B\to C\to 0$ of abelian groups expresses the fact that $A$ is a subgroup of $B$ and $B/A\cong C$. 


My answer is not so different from Emerton's, but it's mainly an answer to the poster rather than just the question. I think often it's often because of taste and experience that people use commutative diagrams. I find in my own research I'm often checking that diagrams commute or checking that sequences are exact, by computing with the actual underlying formulas. However I find it easier to state (and remember) things with the diagrams. In terms of intuition diagrams give a different point of view and so may lead to different ideas and understandings of things (which is a good thing). I have you a bigger commutative diagram (say like a cube) saying the diagram is commutative, is easier to understand and phrase. Rewording this in terms of equations would probably make it less clear. A very simple diagram is $A\rightarrow B$, do you feel this is a useful diagram, or would you prefer to have maps only described by equations? That might be hard if you were dealing with abelian categories. In terms of the example of tensor products, I also can understand the actual object better as a more concrete thing, but when you want to construct maps out of it, sometimes it's easier to use its "universal property" (ie that bilinear maps out of the product factor uniquely through it). Once I was siting in on a course on representation theory by a colleague. He was checking that some map existed (and was unique) by checking element by element, but actually all he really had to do was to check that some other map was bilinear. In this case it would have been much easier to use the universal property of the tensor product. On the other hand I do agree that as an object itself it's easier to understand the tensor product as linear combinations modulo some equivalence relation. 


There are at least two advantages of the approach via universal morphisms. One is that it relates to the idea that one studies a particular object by its relation to all other objects. The other is that it allows for generalisations: an example is that from tensor products of abelian groups to the nonabelian tensor product of groups which act on each other. In fact this suggest another advantage: replace a complicated map by a simpler one, in this case a morphism. In particular a morphism of abelian groups has a kernel, whereas a biadditive map does not have that notion. However the isomorphism theorems in group theory arose from the need to systematise facts on particular examples, and these are the, or a, main delight of basic group theory. I confess I did not get round to doing calculations in group theory till I was 50  I had previously missed out on this. 


(too many characters for me to leave a comment on Emerton's answer): I was asking why are elementary theorems phrased in this language and why is this language used at all. The question was caused by my nervousness when this language appears and by the following excerpt from a review of J. S. Milne's Etale cohomology by Spencer Bloch: ... in leafing through the collected works of Weil (who in some sense started it all) I am unable to find a single exact sequence or commutative diagram. The reader is invited to compare this with the work under review (or indeed with any of the published work of either author or reviewer). It would be interesting to see more clearly the shift in mathematical philosophy which must underlie this shift in notation. 

