ash is a small implementation of the Bourne shell but with some additional features/extensions taken from ksh/bash. It's part of the busybox tools and very poorly documented (google for its documentation and all you get is posts from other people asking where to find the documentation), although most of the documentation for the Debian Ash (dash) shell is applicable.
I tend to write scripts in ksh these days as it's on all of the system types I use. It's largely bourne compatible but has some extra features and different behaviour in some instances (such as not spawning a subshell for a while loop).
Anyway, if you're going to learn a shell, learn Bourne. If you insist on learning Bash then do try and keep a handle on what is a Bash extensions versus standard Bourne and don't complain that the up arrow is broken the first time you use a pure Bourne shell!
(for my interactive sessions, I favour zsh - so many choices!)