how to get the current chapter name, section name, subsection name, etc?

With this solution you can still use the starred versions and optional arguments of \chapter et al.

\documentclass{book}
\let\Chaptermark\chaptermark
\def\chaptermark#1{\def\Chaptername{#1}\Chaptermark{#1}}
\let\Sectionmark\sectionmark
\def\sectionmark#1{\def\Sectionname{#1}\Sectionmark{#1}}
\let\Subsectionmark\subsectionmark
\def\subsectionmark#1{\def\Subsectionname{#1}\Subsectionmark{#1}}
\let\Subsubsectionmark\subsubsectionmark
\def\subsubsectionmark#1{\def\Subsubsectionname{#1}\Subsubsectionmark{#1}}

\begin{document}
\chapter{First chapter}\label{ch:first}
This is chapter~\ref{ch:first} with title ``\Chaptername''.
\section{First section}\label{sec:first}
This is section~\ref{sec:first} with title ``\Sectionname''.
\subsection{The first subsection}\label{subsec:first}
This is subsection~\ref{subsec:first} with title ``\Subsectionname''.
\subsubsection{Last subsubsection}\label{subsubsec:first}
This is subsubsection~\ref{subsubsec:first} with title ``\Subsubsectionname''.
\end{document}

In terms of section titles, the nameref package is ideal for this. It provides \nameref{<label>} that returns the sectional title associated with the label:

\documentclass{article}
\usepackage{nameref}% http://ctan.org/pkg/nameref
\begin{document}
\section{First section}\label{first}
This is section~\ref{first} with title \nameref{first}.
\subsection{Second subsection}\label{second}
This is subsection~\ref{second} with title \nameref{second}.
\subsubsection{Last subsubsection}\label{third}
This is subsubsection~\ref{third} with title \nameref{third}.
\end{document}

hyperref provides similar functionality, since it loads nameref. As such, they work together without problem, allowing hyperlinked titles. memoir does something similar to store the names/titles of sectional units.

This could be expanded to include the capture of names/titles associated with other environments or structures as well.

Here is one solution but is not optimal. It simply renew's the old sectioning commands and have the new ones update the currentxxx macros.

\begin{document}

\newcommand{\currentchapter}{}
\let\oldchapter\chapter
\renewcommand{\chapter}[1]{\oldchapter{#1}\renewcommand{\currentchapter}{#1}}

\newcommand{\currentsection}{}
\let\oldsection\section
\renewcommand{\section}[1]{\oldsection{#1}\renewcommand{\currentsection}{#1}}

\newcommand{\currentsubsection}{}
\let\oldsubsection\subsection
\renewcommand{\subsection}[1]{\oldsubsection{#1}\renewcommand{\currentsubsection}{#1}}

\newcommand{\currentsubsubsection}{}
\let\oldsubsubsection\subsubsection
\renewcommand{\subsubsection}[1]{\oldsubsubsection{#1}\renewcommand{\currentsubsubsection}{#1}}


\chapter{chapter test}
\section{section test}
\subsection{subsection test}

\currentchapter\\
\currentsection\\
\currentsubsection\\

\end{document}