What's so special about $1$-categories? I have been pretty thoroughly convinced for some time now that, when thinking about mathematics, one really should be thinking 'categorically', that is, one should always be thinking of the morphisms between objects instead of just the objects themselves.  You might say that, as I have mathematically matured, my tendency has been to think more at the level of $1$-categories instead of at the level of $0$-categories.
Phrasing it like that made me wonder:  is this just the first step?  Should I 'really' be thinking not about morphisms themselves, but about morphisms between morphisms?  Indeed, of the couple of examples I can think of off the top of my head where such $2$-morphisms arise quite naturally, it does seem to be the 'right' thing to do to think about morphisms up to isomorphism (the best example I have in mind is the difference between isomorphism and equivalence of categories, the former not being quite as useful).
Of course, once you've decided to think $2$-categorically, why not think $3$-categorically or $4$-categorically, or hell, why not $\infty$-categorically?
I apologize for this being a subjective question, but:  To what extent should I begin to train myself to think in terms of higher category theory, or, is there indeed something special occurring at the 'normal' level of $1$-categories that makes this level in particular the 'right' way to think about things?
 A: You should increase your category level if you think it'll help you understand something you're thinking about. Otherwise, don't.
To the extent that there's something special about $1$-categories, it might be that on the one hand it's a big step up from $0$-categories and on the other hand it's a low enough category level that you don't need to worry about coherence conditions. Already at the $2$-categorical level an important distinction arises between weak and strict $2$-categories (when I say "$2$-category" I mean the weak thing by default because the alternative does not generalize to higher values of $2$) that takes some time to appreciate, although you can attempt to ignore it by strictifying, and at the $3$-categorical level you can't even always strictify.. 
A: Taking the homotopical point of view, a 1-category is a space where an imbedding of a more than one dimensional cell is determined by the boundary. On the one hand this makes it easier to calculate and make concrete statements, on the other it is often rather limiting and unnatural.
A good example of this is the derived category, which can be seen as the homotopy category of a higher category.
