In this proof for the rationals being countably infinite, is the enumeration strategy important? 
During some set theory exercises I came across the proof that the rationals are countably infinite. It used what appears to be a common proof where all the pairs are listed and a zig-zagging path is taking through them, starting with $(1,1), (1,2), (2,1)...$ and so forth.  
Of course I am tasked with proving all kinds of related things and my first snag is the following: Is the path taken, or maybe better put, is the enumeration strategy important?
I'm guessing it is, if you would just list all pairs $\sum_{i=1} (1,i)$ first, you would never reach $(2,1), (2,2)...(3,1)...$ and such. From a computational perspective, this makes sense to me. Intuitively, not so much, since we are already dealing with infinites.
But the way my textbook does it, you have a diagonal of pairs, starting with $(1,1)$ that grows and as such all the rows get even "exposure". Is this reasoning correct?
Note, my textbook doesn't count 0 as a natural number.

For the traditional bijection with the positive integers you only get one free trip to infinity. The bijection you proposed requires infinitely many trips to infinity. In set theory ordered mappings beyond infinity are associated with "ordinal" numbers.
Ordinal $\omega$ maps to {1 .. $\infty$}. The sequence {2 .. $\infty$, 1} would be $\omega+1$. The sequence {2, 4, 6, 8, .. $\infty$, 1, 3, 5, 7, $\infty$} makes two trips to infinity and is mapped to $\omega + \omega = 2\cdot\omega = 2\omega$. The mapping you described is represented by $\omega \cdot\omega = \omega^2$.
If you know that $\omega^2$ is countable then you've proved the rationals are countable. But that requires knowledge beyond a typical countability proof.
