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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.

beginscope[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$3$;
 endscope
 beginscope[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);
 endscope

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.

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
usetikzlibraryshapes.geometric,calc
newcounterhexi
begindocument
begintikzpicture[pics/hexi/.style=code=stepcounterhexi
node[draw,regular polygon,regular polygon sides=6,minimum width=2cm]
(hexi-numbervaluehexi) #1;
foreach Corner in 1,...,6
ifoddCorner
 draw[fill=black] (hexi-numbervaluehexi.corner Corner) circle[radius=1.5pt];
else
 draw[fill=white] (hexi-numbervaluehexi.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 pichexi=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;
endtikzpicture
enddocument

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
usetikzlibrarydecorations.pathreplacing,calc,shapes.geometric
tikzstylewith 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] ($(tikzinputsegmentfirst)!1mm!45:(tikzinputsegmentlast)$) -- ($(tikzinputsegmentlast)!1mm!-45:(tikzinputsegmentfirst)$);
 
 
 
]
begindocument
 begintikzpicture
 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;
 endtikzpicture
enddocument