This is not a complete answer to your question. It is just a good example to start with knowing that there are many approaches to your problem.
\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{matrix}
\begin{document}
\begin{tikzpicture}[->,>=stealth]
\matrix (dag) [matrix of nodes,%
nodes={rectangle,draw},
column sep={2cm,between origins},
row sep={2cm,between origins},
ampersand replacement=\&] {
\& |(1)| Start \& \\
|(11)| A \& |(12)| B \& |(13)| C \\
\& |(21)| End \& \\
};
\draw (1) to (11);
\draw (1) to (12);
\draw (1) to (13);
\draw (11) to (12);
\draw (12) to (13);
\draw (11) to (21);
\draw (12) to (21);
\draw (13) to (21);
\end{tikzpicture}
\end{document}

Edit
I am not the TikZpert here in the slightest, but I spent some time converting your hand drawing into a simple code. Please, study it well since do-it-for-me questions are the most hated ones here :)
.
\documentclass[12pt]{article}
\usepackage{tikz}
\usetikzlibrary{matrix}
\begin{document}
\begin{figure}
\begin{tikzpicture}[->,>=stealth,]
\matrix (dag) [matrix of nodes,%
nodes={rectangle,draw},
column sep={3ex},
row sep={10ex,between origins},
ampersand replacement=\&] {
\& |(01)| $\sigma$-rigid \& \\
|(11)| strongly $\sigma$-symmetric \& |(12)| strongly symmetric \& |(13)| symmetric \\
|(21)| strongly $\sigma$-skew reversible \& |(22)| strongly reversible \& |(23)| reversible \\
|(31)| strongly $\sigma$-IFP \& |(32)| strongly IFP \& |(33)| IFP \\
\& |(41)| Abelian \& \\
};
\draw (01.south west) -- (11.north east); \draw (01) -- (12); \draw (01.south east) -- (13.north west);
\draw[<->] (11) -- node[midway,right]{text} (21); \draw (21) -- (31); \draw (31.south east) -- (41.north west);
\draw (12) -- (22); \draw (22) to (32); \draw (32) -- (41);
\draw[<->] (13) -- (23); \draw (23) -- (33); \draw (33.south west) -- (41.north east);
\draw (11) -- (12); \draw (12) -- (13); \draw (12) -- (13);
\draw (21) -- (22); \draw (22) -- (23); \draw (22) -- (23);
\draw (31) -- node[above] {dummy} (32); \draw (32) -- (33); \draw (32) -- (33);
\end{tikzpicture}
\end{figure}
\end{document}
