Minimal syntax highlighting for Tex File, with only math expressions, section-structures and begin/end{environment}

Question

My Vim is very slow at rendering the syntax highlighting of LaTeX documents using the default syntax file to be found in c:\vim\vim80\syntax\tex.vim. The problem is partially solved in this post, yet it is hard to write LaTeX document with no visual clue. What is a good way to start writing a minimal syntax file by myself? By minimal, I am expecting to have:

1. Highlighting math expressions enclosed by $, \[ and \]; also, those Tex specific commands like \pi.
2. Sections, usually in form of \section{Section Name} and \subsection{Subsection Name}.

3. Environments, usually in form:

   \begin{environment}

   \end{environment}