Q:

Space before and after a character/symbol

Suppose I want to define \newcommand{\versesep}{\textbullet} to be used to separate, say, Bible verses. What is the suggested method to define the horizontal space before and after the character? 

A:

Something like this??  Unlike the \kern alternative, the \hspace will be absorbed at margins, allowing proper alignment/flushing.  The particular numbers for the gap can be chosen by the OP.
Thus use of minus (or plus) in defining the gapsize allows for some slop to be built in to the gap, which can be useful especially for narrow column documents.
\documentclass{article}
\def\versegap{\hspace{1pt minus.5pt}}
\newcommand\versesep{\versegap\textbullet\versegap}
\textwidth3in
\begin{document}
God created the heavens and the earth,\versesep the earth was a formless void and

xxxGod created the heavens and the earth,\versesep the earth was a formless void and

xxxxxxGod created the heavens and the earth,\versesep the earth was a formless void and
\end{document}

If you wanted to avoid a linebreak following the bullet, then this variation could be used:
\documentclass{article}
\def\versegap{\hspace{1pt minus.5pt}}
\def\versekern{\kern\glueexpr1pt minus.5pt\relax}
\newcommand\versesep{\versegap\textbullet\versekern\nobreak}
\textwidth3in
\begin{document}
God created the heavens and the earth,\versesep the earth was a formless void and

xxxGod created the heavens and the earth,\versesep the earth was a formless void and

xxxxxxGod created the heavens and the earth,\versesep the earth was a formless void and
\end{document}

