How to remove chapter numbering without removing it from table of contents

By adding \setcounter{chapter}{n} before the nth chapter seems to fix the numbering of subsections issue. Also though, you have to reset the numbering of sections. An easy way to do this would to just define a new command. For example:

\documentclass{report}
\newcommand{\mychapter}[2]{
    \setcounter{chapter}{#1}
    \setcounter{section}{0}
    \chapter*{#2}
    \addcontentsline{toc}{chapter}{#2}
}
\begin{document}
\tableofcontents
\mychapter{0}{Acknowledgments}
\mychapter{1}{Introduction}
\section{Introduction}
\section{Further Introduction}
\mychapter{2}{Experiments}
\section{Experiment One}
\section{Experiment Two}
\end{document}

You can do something as is done in here. Use code similar to the following:

\documentclass{report}
\begin{document}
\tableofcontents
\chapter*{Acknowledgments}
\addcontentsline{toc}{chapter}{Acknowledgments}
\chapter*{Introduction}
\addcontentsline{toc}{chapter}{Introduction}
\chapter{Experiments}
\chapter{Conclusion}
\end{document}

The idea is to use \chapter*{} for unnumbered chapters and then add them to the toc explicitly with the \addcontentsline. Which gives this result:

If I read that correctly, you want chapters to have numbers. These numbers should be shown in the ToC. But the numbers should not be printed in the chapter heading. This can be achieved with the titlesec package:

\documentclass{report}
\usepackage{titlesec}

\titleformat{\chapter}
  {\Large\bfseries} % format
  {}                % label
  {0pt}             % sep
  {\huge}           % before-code

\begin{document}
\tableofcontents
\chapter{Acknowledgments}
\chapter{Introduction}
\section{Introduction}
\section{Further Introduction}
\chapter{Experiments}
\section{Experiment One}
\section{Experiment Two}
\end{document}

The ToC then looks like

while the chapter heads in the text look like