All condescension aside, my first thought was that, in fact, category theory is an incredibly useful tool and language. As such, many of us want to read CWM so that we can understand various constructions in other fields (for instance the connection between monadicity and descent, or the phrasing of various homotopy theory ideas as coends, not to mention just basic pullbacks, pushforwards, colimits, and so on). So it is in fact relevant WHY you want to read it.
As an undergraduate, I started reading CWM, with minimal success. The idea being primarily that, as you say, I had very few examples. I thought the notion of a group as a category with one element was rather neat, but I couldn't really understand adjunctions, over(under)-categories, colimits or some of the other real meat of category theory in any deep, meaningful fashion until I began to have some examples to apply.
In my opinion, it is not fruitful to read CWM straight up. It's like drinking straight liquor. You might get really plastered (or in this analogy, excited about all the esoteric looking notation and words like monad, dinatural transformation, 2-category) but the next day you'll realize you didn't really accomplish anything.
What is the rush? Don't read CWM. Read Hatcher's Algebraic Topology, read Dummit and Foote, read whatever the standard texts are in differential geometry, or lie groups, or something like that. Then, you will see that category theory is a lovely generalization of all the nice examples you've come to know and love, and you can build on that.
Best Answer
Some analysts (in a wide sense) write $\langle x,y\rangle$ (angle brackets), for $x$ and $y$ elements of a same set $X$, to denote the ordered pair element of $X\times X$. A more classical (to me) notation is $(x,y)$ (parenthesis), but, if for example $X=\mathbb R$ and $x\leqslant y$, the notation $(x,y)$ may refer to the open interval $\{z\in\mathbb R\mid x<z<y\}$. Hence the bracket notation might have been designed as a way to avoid the confusion. Bourbaki use the notation $]x,y[$ for open intervals and $[x,y]$ for segments. This notation, of frequent use in the mathematical literature written in French (and in others), removes the risk of confusion mentioned above.
I do not know how useful brackets are for objects like $\langle\mathbb Z,+\rangle$, since the objects inside the brackets are of a different nature.