AtBeginEndPackage

First, obtain the style file from here: http://www-h.eng.cam.ac.uk/help/tpl/textprocessing/atbeginend.sty

Now install the style file (also see Cancel_Package#Using_Cancel_Package_in_LyX page on this wiki) by putting it into ~/Library/texmf/tex/latex (if you are on a Mac).

You may need to create this directory:

$ mkdir -p ~/Library/texmf/tex/latex

Now you should be able to fire up LyX and use it.

Open a document, click Document > Settings..., and in the preamble, put

\renewcommand{\labelenumi}{\LARGE \textbf{\arabic{enumi}.}}
\renewcommand{\labelenumii}{\normalsize \textbf{\arabic{enumi}.\arabic{enumii}.}}
\renewcommand{\labelenumiii}{\normalsize \textbf{\arabic{enumi}.\arabic{enumii}.\arabic{enumiii}.}}
%% the following according to http://www.eng.cam.ac.uk/help/tpl/textprocessing/squeeze.html
%% add some space:
\usepackage{atbeginend}
\AfterBegin{itemize}{%
  \addtolength{\itemsep}{0.3\baselineskip}%
}
\AfterBegin{enumerate}{%
  \addtolength{\itemsep}{0.3\baselineskip}%
}

or alternatively, to use letters for sub-items:

\renewcommand{\labelenumi}{\LARGE \textbf{\alph{enumi}.}}
\renewcommand{\labelenumii}{\normalsize \textbf{\alph{enumii}.}}
\renewcommand{\labelenumiii}{\normalsize \textbf{\alph{enumiii}.}}
%% the following according to http://www.eng.cam.ac.uk/help/tpl/textprocessing/squeeze.html
%% add some space:
\usepackage{atbeginend}
\AfterBegin{itemize}{%
  \addtolength{\itemsep}{0.3\baselineskip}%
}
\AfterBegin{enumerate}{%
  \addtolength{\itemsep}{0.3\baselineskip}%
}