Use
\renewcommand{\labelenumii}{\theenumii}
\renewcommand{\theenumii}{\theenumi.\arabic{enumii}.}
to change the way nested enumerations are set (without brackets ()) and in a sub-enumeration form that is \arabic:

\documentclass{article}
\renewcommand{\labelenumii}{\theenumii}
\renewcommand{\theenumii}{\theenumi.\arabic{enumii}.}
\begin{document}
\begin{enumerate}
\item First
\begin{enumerate}
\item Second
\item Third
\end{enumerate}
\item Fourth
\end{enumerate}
\end{document}
The enumitem is actually the de-facto list processing/setting package. Here's such a take on your enumeration:
\documentclass{article}
\usepackage{enumitem}% http://ctan.org/pkg/enumitem
\begin{document}
\begin{enumerate}
\item First
\begin{enumerate}[label*=\arabic*.]
\item Second
\item Third
\end{enumerate}
\item Fourth
\end{enumerate}
\end{document}
label* appends the suggested key-value to the parent label. Of course, global setting is also possible.
\label{}and\ref{}items in your list, the first option gives a bad\ref(the \enumi part appears twice). The enumitem method avoids this. Also, FYI: the global options look like:\setlist[enumerate,1]{label=\arabic*.} \setlist[enumerate,2]{label*=\arabic*.}– jessexknight Oct 17 '18 at 18:11