I think I'd want to deal with partially defined functions $f: [n] \to [n']$, with the property that if $F$ is a face of $\Delta$, then $f(F)$ is a face of $\Delta'$. The linear extension of such an $f$ is a linear map from $K^n \to K^{n'}$, taking the $i$th basis vector to the $f(i)$th, or to zero if $f(i)$ is not defined.
The Stanley-Reisner ideal is the functions vanishing on the union $X_\Delta \subseteq K^n$ of coordinate spaces $\bigcup_{F\in\Delta} K^F$, where $K^F$ denotes the linear span of {the $i$th basis vector : $i\in F$}. (Does that answer your second question?) Then the linear extension of $f$ takes $X_\Delta$ into $X_{\Delta'}$.