As I explained in the comments, there is some method in the way biblatex formats the titles for certain entry types: The titles of works that stand on their own (@books, @collections, entire journals, ...; basically anything you can read on the spine or the cover of a book when you stroll through a library) are in italics, while dependent works that constitute only a part of a whole (@inbook, @incollection, @article, ...; these are mainly titles you can only find out about when you open a book or volume and consult the table of contents) have their titles formatted with quotation marks.
You can find an explanation of the default settings in Remove Quotation Marks from Style. You probably like to try
\DeclareFieldFormat[online]{title}{\mkbibquote{#1\isdot}}
if you really want only book titles in italics and the rest in quotation marks, you could try
\DeclareFieldFormat*{title}{\mkbibquote{#1\isdot}}
\DeclareFieldFormat[book]{title}{\mkbibemph{#1\isdot}}
For German-style quotation marks it should be enough to load babel and pass ngerman as loading option and then load csquotes for language-dependent quotation marks. Of course that is assuming that the rest of your document is in German, but I think that is a fair assumption if you are looking for German quotation marks.
\RequirePackage{filecontents}
\begin{filecontents*}{\jobname.bib}
@book{darwin,
author = {Charles Darwin},
publisher = {Cambridge University Press},
title = {The Correspondence of Charles Darwin, 1821--1836},
year = {1985},
}
@electronic{wade,
title = {Darwin, Ahead of His Time, Is Still Influential},
author = {Nicholas Wade},
date = {2009-02-09},
url = {https://www.nytimes.com/2009/02/10/science/10evolution.html},
urldate = {2018-10-08},
}
\end{filecontents*}
\documentclass[ngerman]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{babel}
\usepackage{csquotes}
\usepackage[style=verbose]{biblatex}
\DeclareFieldFormat[online]{title}{\mkbibquote{#1\isdot}}
\addbibresource{\jobname.bib}
\begin{document}
Lorem \autocite{darwin} ipsum \autocite{wade}
\printbibliography
\end{document}

@inbook,@incollection,@articles, ...) have their titles in quotation marks. Do you explicitly want all titles in quotation marks? – moewe Dec 11 '18 at 09:17.bibfile is "The Correspondence of Charles Darwin, 1821-1836") and not outside it? – moewe Dec 11 '18 at 09:31