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I am able of doing the hexagons and the rectangle, as well as all the nodes and so on. My problem is if there is an easy way to draw the blue arrows of the image, knowing that I have given to tikzpicture the coordinate of each hexagon.

begin{scope}[xshift=-1.5cm,yshift=7.83cm]

    node[draw,circle,inner sep=2.5pt,minimum size=2pt,fill=black] (A) at (0:1cm) {};

    node[draw,circle,inner sep=2.5pt,minimum size=2pt] (B) at (60:1cm) {};

    node[draw,circle,inner sep=2.5pt,minimum size=2pt,fill=black] (C) at (120:1cm) {};

    node[draw,circle,inner sep=2.5pt,minimum size=2pt] (D) at (180:1cm) {};

    node[draw,circle,inner sep=2.5pt,minimum size=2pt,fill=black] (E) at (240:1cm) {};

    node[draw,circle,inner sep=2.5pt,minimum size=2pt] (F) at (300:1cm) {};

    draw[thick] (A)--(B);

    draw[thick] (B)--(C);

    draw[thick] (C)--(D);

    draw[thick] (D)--(E);

    draw[thick] (E)--(F);

    draw[thick] (F)--(A);

    node at (0:0cm) {scriptsize$3$};

    end{scope}

    begin{scope}[xshift=-1.5cm,yshift=6.09cm]

    node[draw,circle,inner sep=2.5pt,minimum size=2pt,fill=black] (A) at (0:1cm) {};

    node[draw,circle,inner sep=2.5pt,minimum size=2pt] (B) at (60:1cm) {};

    node[draw,circle,inner sep=2.5pt,minimum size=2pt,fill=black] (C) at (120:1cm) {};

    node[draw,circle,inner sep=2.5pt,minimum size=2pt] (D) at (180:1cm) {};

    node[draw,circle,inner sep=2.5pt,minimum size=2pt,fill=black] (E) at (240:1cm) {};

    node[draw,circle,inner sep=2.5pt,minimum size=2pt] (F) at (300:1cm) {};

    draw[thick] (A)--(B);

    draw[thick] (B)--(C);

    draw[thick] (C)--(D);

    draw[thick] (D)--(E);

    draw[thick] (E)--(F);

    draw[thick] (F)--(A);

    node at (0:0cm) {scriptsize$4$};

    coordinate (1c) at (280:0.7cm);

    end{scope}

Above there is an example of how I programmed two adjacent hexagons. I programmed one and then I shifted the reference frame to have the second one below. With "coordinate" I save a point so that in the end I can draw the red rectangle. Based on this way of programming, How can I add the blue arrows? There should be a way to put the arrows parallel to the line joining the two nodes, and then maybe with decorate I can add the number of >> that I need.
Any suggestion?
Thank you, I apologize if this is not the best way to draw this tiling but it is the most versatile for what I need to do, so I would like not to change it, but I am interested in how to add parallel lines joining (or pointing) to two nodes.

Your code is far from optimal, so I'll not reproduce it here. I only show you how to add the blue arrow next to an edge, as you asked, by creating a style with arrows.

EDIT: I added also a closepath code to with arrows style in a way to be able to use it with regular polygon nodes (shown already in the @marmot's answer).

documentclass[tikz,border=7pt]{standalone}

usetikzlibrary{decorations.pathreplacing,calc,shapes.geometric}

tikzstyle{with arrows}=[

    postaction={decorate,

      decoration={show path construction,

        lineto code={

          draw [blue,-latex] ($(tikzinputsegmentfirst)!1mm!45:(tikzinputsegmentlast)$) -- ($(tikzinputsegmentlast)!1mm!-45:(tikzinputsegmentfirst)$);

        },

        closepath code={

          draw [blue,-latex] ($(tikzinputsegmentfirst)!1mm!45:(tikzinputsegmentlast)$) -- ($(tikzinputsegmentlast)!1mm!-45:(tikzinputsegmentfirst)$);

        }

      }

    }

]

begin{document}

  begin{tikzpicture}

    draw[with arrows] (0,0) -- (1,1) node[scale=2]{.} -- (2,0);

    node[regular polygon,regular polygon sides=6,minimum width=2cm,draw,with arrows] at (3,1) {1};

  end{tikzpicture}

end{document}

It is not too difficult to draw such a thing in loops. pics may further help to avoid repetition. I did not understand your numbering scheme so you will have to modify evaluate=Y as Z using {int(mod(33-Y-X,5)+1)} to match your prescription.

documentclass[tikz,border=3.14mm]{standalone}

usetikzlibrary{shapes.geometric,calc}

newcounter{hexi}

begin{document}

begin{tikzpicture}[pics/hexi/.style={code={stepcounter{hexi}

node[draw,regular polygon,regular polygon sides=6,minimum width=2cm]

(hexi-numbervalue{hexi}) {#1};

foreach Corner in {1,...,6}

{ifoddCorner

 draw[fill=black] (hexi-numbervalue{hexi}.corner Corner) circle[radius=1.5pt];

else

 draw[fill=white] (hexi-numbervalue{hexi}.corner Corner) circle[radius=1.5pt];

fi}

}},bullet/.style={circle,fill,inner sep=0.5pt}]

%

 clip (0,1) rectangle (9.8,6.5);

% draw the hexagons 

 path foreach X in {1,...,6} {

 foreach Y [evaluate=Y as Z using {int(mod(33-Y-X,5)+1)}] in {1,...,4} { ifoddX 

  ({X*(1+cos(60))},{Y*(2*sin(60))})

 else

  ({X*(1+cos(60))},{Y*(2*sin(60))-sin(60)})

 fi pic{hexi=Z}}};

% draw the blue arrows

 foreach X in {7,9,10,11,14}

 {foreach Y [remember=Y as LastY (initially 6)]in {1,...,6}

 {draw[blue,-latex,shorten >=2pt,shorten <=2pt] 

  ($(hexi-X.corner LastY)!0.1!(hexi-X.center)$)

  -- ($(hexi-X.corner Y)!0.1!(hexi-X.center)$);}}

% draw the red contour

draw[red] ([yshift=-0.3cm]hexi-3.center) node[bullet]{}

  -- ([yshift=-0.3cm]hexi-6.center) node[bullet]{}

  -- ([yshift=-0.3cm]hexi-18.center) node[bullet]{}

  -- ([yshift=-0.3cm]hexi-16.center) node[bullet]{} -- cycle;

end{tikzpicture}

end{document}