Use counter before its value is set

Welcome to TeX.SE. This is a job for the totcount package. You need to compile twice, since totcount uses the .aux file to store the last value assigned to the counter in the document.

Note that I use \value{enumi} instead of \thenumi in the \setcounter call in case \thenumi doesn't expand to an integer denotation (\thenumi might use \roman or \alph formatting, etc.).

\documentclass{article}
\usepackage{totcount}

\begin{document}

\newtotcounter{howmany}
There are \total{howmany}~items in this list:
\begin{enumerate}
  \item one
  \item two
  \item three
\setcounter{howmany}{\value{enumi}}
\end{enumerate}

\end{document}

If there is already a \newcounter command for the counter in question and you don't want to, or cannot change this command into a \newtotcounter call, you can register the counter with the totcount package using \regtotcounter{countername} (this can be useful if the counter is defined by a package).

This also works with non-decimal labels

This technique works as well if you use non-decimal label formatting:

\documentclass{article}
\usepackage{totcount}

\renewcommand{\theenumi}{\alph{enumi}}

\begin{document}

\newtotcounter{howmany}
There are \total{howmany}~items in this list; their labels are
\ref{first-item}, \ref{second-item}, and \ref{third-item}.
\begin{enumerate}
  \item \label{first-item}one
  \item \label{second-item}two
  \item \label{third-item}three
\setcounter{howmany}{\value{enumi}}
\end{enumerate}

\end{document}

You can wrap your list to be counted in an environment:

\documentclass{article}

\makeatletter
\newenvironment{countitems}[1]
 {% #1 is the label for referring to the count
  \def\countitems@label{#1}%
 }
 {%
  \edef\@currentlabel{\arabic{enum\romannumeral\numexpr\@enumdepth+1}}%
  \label{\countitems@label}%
  \ignorespacesafterend
 }
\makeatother

\begin{document}

The following list has \ref{firstlist} items
\begin{countitems}{firstlist}
\begin{enumerate}
\item one
\item two
\item three
  \begin{countitems}{innerlist}
  \begin{enumerate}
  \item A
  \item B
  \end{enumerate}
  \end{countitems}
\end{enumerate}
\end{countitems}
The inner list has \ref{innerlist} items.

\end{document}

Alternatively, you can use a command after the relevant \end{enumerate}

\documentclass{article}

\makeatletter
\newcommand{\countitems}[1]{%
  \edef\@currentlabel{\arabic{enum\romannumeral\numexpr\@enumdepth+1}}%
  \label{#1}%
  \ignorespaces
 }
\makeatother

\begin{document}

The following list has \ref{firstlist} items
\begin{enumerate}
\item one
\item two
\item three
  \begin{enumerate}
  \item A
  \item B
  \end{enumerate}\countitems{innerlist}
\end{enumerate}\countitems{firstlist}
The inner list has \ref{innerlist} items.

\end{document}