Drawing a Grid/Tikz image

Using a loop here is probably over-killing, but anyway, here is a modification of your code. Single draw command to do this ;)

\documentclass[tikz,border=7pt]{standalone}
\begin{document}
  \tikz
    \draw[nodes={inner sep=1pt},color=gray]
      (0,0) grid[shift={(-.5,-.5)}] (4,4)
      foreach[count=\i] \x/\y/\m in {2/3/a,2/2/b,3/3/c,}{(\x,\y) node(n\i) {\m}}
      (n1) edge[red,-latex] (n3)
      (n2) edge[red,-latex] (n3)
    ;
\end{document}

I would draw your diagram like this

\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}[>=stealth]
\foreach \i in {0,1,2,3.5,4.5,5.5,7,8} {
    \draw[gray,very thin] (0,\i) -- (8.5,\i);
    \draw[gray,very thin] (\i,0) -- (\i,8.5);
}
\draw[<->] (8.5,0) node[below] {$n$} -- (0,0) node[below left] {$(1,1)$} -- (0,8.5) node[left] {$n$};
\draw[->] (2,-.5) -- (3,-.5) node[above,near start] {$r$};
\foreach \i in {2.75,6.25} {
    \foreach \j in {.5,1.5,4,5,7.5} {
        \node at (\i,\j) {$\cdots$};
        \node at (\j,\i) {$\vdots$};
    }
    \foreach \j in {2.75,6.25}
        \node at (\i,\j) {\reflectbox{$\ddots$}};
}
\begin{scope}[every node/.style={
    circle,draw,
    minimum size=.8cm,
    align=center,
    font=\footnotesize,
    inner sep=0pt
}]
    \node at (4,4) (11) {$\scriptstyle l-1$\\[-1ex]$\scriptstyle r-1$};
    \node at (5,4) (21) {$\scriptstyle l-1$\\[-1ex]$\scriptstyle r$};
    \node at (4,5) (12) {$\scriptstyle l$\\[-1ex]$\scriptstyle r-1$};
    \node at (5,5) (22) {$\scriptstyle l$\\[-1ex]$\scriptstyle r$};
\end{scope}
\draw[thick,red,->] (11) -- (22);
\draw[thick,red,->] (12) to[bend left=45] (22);
\draw[thick,red,->] (21) to[bend right=45] (22);
\end{tikzpicture}
\end{document}