Typically you can write your own environment that wraps around the one you want to change. For example
\newenvironment{quotingalt}
{% \begin{quotingalt}
\large% Change font size
\begin{quote}%
}{% \end{quotingalt}
\end{quote}%
}
In fact, this is how many of the default environments in the standard classes are written. For example, consider the quote environment from article.cls:
\newenvironment{quote}
{\list{}{\rightmargin\leftmargin}%
\item\relax}
{\endlist}
The above code initiates the list environment with \list and ends it with \endlist to form quote.
Since the environment has some scope, changes are localized so you don't have to worry about resetting them. The environ package allows a more intuitive command-like interface to environments, providing \BODY for the grabbed content inside it:
\usepackage{environ}% http://ctan.org/pkg/environ
\NewEnviron{quotingalt}{%
\large% Change the font size
\begin{quote}\BODY\end{quote}%
}
Alternative approaches include keeping the original name and modifying it using a patch (with etoolbox and friends) or a complete redefinition with your style added.