\documentclass[10pt]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.15}
\usepackage{mathrsfs}
\usetikzlibrary{arrows}
\pagestyle{empty}

\begin{document}

% Define colors
\definecolor{bqqqsq}{rgb}{0.69,0.00,0.13} % dark red
\definecolor{qqqqff}{rgb}{0.00,0.00,1.00} % blue
\definecolor{rvwvcq}{rgb}{0.08,0.40,0.75} % teal

\begin{tikzpicture}[line cap=round, line join=round, >=triangle 45, scale=1.5]

\begin{axis}[
  x=1.0cm,y=1.0cm,
  axis lines=middle,
  grid=major,
  xmin=-1, xmax=9.3,
  ymin=-0.8, ymax=9.6,
  xtick={0,...,9},
  ytick={0,...,9}
]

% Lines
\draw[very thick] plot[domain=-1:9.3] (\x,{\x});
\draw[very thick, color=qqqqff] plot[domain=-1:9.3] (\x,{\x-1});

% Labels for lines
\node at (3,3.7) {$y = x$};
\node[color=qqqqff] at (4.2,2.2) {$y = x-1$};

% Route.
\draw[very thick,orange] (0,0) -- (2,0) -- (2,1) -- (5,1) -- (5,2) -- (5,3) -- (6,3) 
    -- (6,6) -- (7,6) -- (7,7) -- (8,7) -- (8,8);

\draw[line width=2pt,brown] (0,0) -- (1,0) (6,5) -- (6,6);

% Points
\fill[rvwvcq] (0,0) circle (2.5pt) node[above left] {$(0,0)$};
\fill[rvwvcq] (8,8) circle (2.5pt) node[above left] {$(n,n)$};
\fill[rvwvcq] (6,6) circle (2.5pt) node[above left] {$(k,k)$};

\fill[bqqqsq] (6,5) circle (2.5pt) node[below right] {$(k,k-1)$};
\fill[bqqqsq] (1,0) circle (2.5pt) node[below right] {$(1,0)$};

\end{axis}
\end{tikzpicture}

\end{document}
