There is no notation for the smallest inaccessible cardinal. In fact, there is no notation "agreed" for any large cardinal property. One may abbreviate things like the tree property, or so, but those are usually properties that can occur at small cardinals.
We would usually say something like,
Let $\kappa$ be the least inaccessible cardinal...
The concept, of course, is very coherent. Every non-empty class of ordinals has a least element. Therefore if the class of inaccessible cardinals is non-empty, it has a least element.
As for the issue of $\aleph_0$, large cardinals are often called "strong axioms of infinity". The reason is that you need the axiom of infinity in order to prove that $\aleph_0$ exists, and you need stronger axioms to prove the existence of inaccessible cardinals, weakly compact, measurable, Woodin, supercompact, huge, and so on.
However, when saying inaccessible cardinals we almost always include "uncountable" in the definition, as to avoid the $\aleph_0$ case.