Indenting lines of code in algorithm

Define:

\algdef{SE}[SUBALG]{Indent}{EndIndent}{}{\algorithmicend\ }%
\algtext*{Indent}
\algtext*{EndIndent}

Then in the algorithmic block, write:

\begin{algorithmic}[1]
    \State Outside indent block
    \Indent
         \State Inside indent block
    \EndIndent
\end{algorithmic}

In the following example code I defined two new commands allowing you to change the indentation; simply enclose the desired fragment using \bindent, \eindent; the length \myindent controls the indent amount:

\documentclass{article}
\usepackage{algorithm,algorithmic}
\usepackage{caption}

\newlength\myindent
\setlength\myindent{2em}
\newcommand\bindent{%
  \begingroup
  \setlength{\itemindent}{\myindent}
  \addtolength{\algorithmicindent}{\myindent}
}
\newcommand\eindent{\endgroup}

\begin{document}

\begin{algorithm}[H]
\caption*{my algorithm}
\begin{algorithmic}
    \STATE \textbf{Stage one:} this is stage one
    \bindent
    \FORALL{i}
        \STATE do something
    \ENDFOR
    \eindent
    \STATE \textbf{Stage two:} this is stage two
    \bindent
    \STATE Update the trie: 
    \FORALL{j}
    \STATE do something
    \ENDFOR
    \eindent
\end{algorithmic}
\end{algorithm}

\end{document}

Some comments to the code:

\newlength\myindent % define a new length \myindent
\setlength\myindent{6em} % assign the length 2em to \myindet
\newcommand\bindent{%
  \begingroup % starts a group (to keep changes local)
  \setlength{\itemindent}{\myindent} % set itemindent (algorithmic internally uses a list) to the value of \mylength
  \addtolength{\algorithmicindent}{\myindent} % adds \mylength to the default indentation used by algorithmic
}
\newcommand\eindent{\endgroup} % closes a group

Problem of Gonzalo's answer is with numbering, as numbers are also indented.

Found better and also simpler solution for \algorithmic, the inbuild environment ALC@g

\begin{ALC@g}
   % Indent what you need
\end{ALC@g}

To compare it in same document

\newlength\myindent
\setlength\myindent{2em}
\newcommand\bindent{%
    \begingroup
    \setlength{\itemindent}{\myindent}
    \addtolength{\algorithmicindent}{\myindent}
}
\newcommand\eindent{\endgroup}

\begin{algorithmic}[1]
    \STATE \textbf{Gonsalo's answer}
    \bindent
    \STATE First
    \STATE Second 
    \eindent
    \STATE \textbf{Proposed answer}

    \begin{ALC@g}
        \STATE First
        \STATE Second 
    \end{ALC@g}
    \STATE Something else
\end{algorithmic}

And result

Of course you can include solution for bigger indent with \addtolength and \setlength inside my proposal.