I'm not quite sure what's ›correct‹ and ›incorrect‹ in your perspective, or what exactly you find problematic with the result you're currently getting. Maybe a minimal example would be helpful.
Here's what I do, and what has always (i.e., for about 5 years now) given me output that's correct by my standards. Over time, I've reduced the number of different ways of entering quotes to two:
For short quotes, or for things that are not quotes but just need quotation marks, single or double, I'm using \MakeAutoQuote in the preamble, so all I have to do is type in quotation marks directly, and let csquotes do the rest (the quotation marks effectively trigger \enquote and \enquote* IIRC). I placed », «, ›, ‹ on [alt gr] + [2...5] on my keyboard, which helped a lot here. If I want it followed by a citation, I add plain old \cite{...} which is handled by biblatex.
For quotes that are longer, and that might have to be turned into a display quote in the output, I use \blockcquote[123]{Knuth82}{blabla}. It is, of course, csquotes that will decide if the quote has to be turned into a display one. That said, for texts whose layout I'm free to design myself, I've disabled display quotes completely: \SetBlockThreshold{99}, plus a re-definition of the display quotes' style. They're marked by quotation marks like text quotes, plus their left margin is reduced by \parindent. (that latter aspect is not part of the example, though)...
\documentclass{scrartcl}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[ngerman]{babel}
\usepackage{lmodern,blindtext}
\usepackage[style=authoryear,autocite=footnote]{biblatex}
\usepackage[german=guillemets]{csquotes}
\MakeAutoQuote{»}{«}
\MakeAutoQuote*{›}{‹}
\SetCiteCommand{\autocite}
\begin{document}
»Knuth wrote in his »famous« book that LaTeX is superb«
\blindtext
\blockcquote[pre][post]{cite}{\blindtext etc}.
\end{document}
I don't let cquotes play with my punctutation, I feel more comfortable staying in control of it myself. Unless I have to follow someone else's style guide (or write in English), my order is always:
[quotation] - [closing mark] - [period or other punctuation] - [footnote mark]
Although the csquotes package allows you to define certain characters as "active" quotes, as your sample document does for the " character, the single quote character cannot be so used, since it is a reserved character in TeX.
Since you mention that you are using text from an MSWord document, a better solution in this case would be to encode your source file as UTF-8 and either use the inputenc package (if you are using pdfLaTeX) or a UTF-8 aware engine (XeLaTeX or LuaLaTeX) to compile your document. In this way you can use the "smart" quote characters “,”, ‘, and ’ directly in your source document. MSWord is quite good at automatically converting quotes correctly. If your document doesn't currently use the smart quotes, you can turn that on and then do a global replace for both ' and " and all the quotes will then be correctly changed.
Source with these quotes can then be used in your LaTeX document. In this case, you don't need to use the csquotes package at all (although you can, still if you have a mixture of quotation marks in the source.)
Using pdfLaTeX
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\begin{document}
I say: “These quotes look nice, but ‘if I quote someone else within a quote like
this,’ or ‘like this,’ they don’t.”
\end{document}
Using XeLaTeX or LuaLaTeX
\documentclass{article}
\usepackage{fontspec}
\begin{document}
I say: “These quotes look nice, but ‘if I quote someone else within a quote like
this,’ or ‘like this,’ they don’t.”
\end{document}
? My initial single quote faces the wrong way, latex thinks it's an apostrophe.
Best Answer
Using "inverted" when you start a single quote part should work:
MWE: