How do you end with a single quote followed by a double quote?

You have the following options:

Outer double, inner single: ``\thinspace`\ \ldots\ '\thinspace'' \par
Outer single, inner double: `\thinspace``\ \ldots\ ''\thinspace'

The TeX Book mentions (Chapter 2: Book Printing versus Ordinary Typing, p 5): "There's usually less space preceding a single left quote than there is preceding a double left quote. (Left and right are opposites.)"

So, in your instance,

``This is a quote. `Here is a quote within a quote.'\thinspace''

The csquotes package might be helpful.

For nested quotings just nest \enquote{} like this:

\documentclass[english]{article}
\usepackage{csquotes}

\begin{document}
\enquote{outer and \enquote{inner}}
\end{document}

It is also very useful for non-english quotation marks:

\documentclass[naustrian,german]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{babel}
\usepackage[babel]{csquotes}

\begin{document}
\enquote{außen und \enquote{innen}}
\end{document}

\documentclass{article}
\begin{document}
``` '''     % misinterpretation by (La)TeX
``{}` '{}'' % good interpretation (but bad spaces) 
\end{document}