Here is a way to achieve what you want by defining the triangles in loops. It is possible to increase the size of the drawing by just modifying the dimensions and adding steps in the loops.

documentclass{standalone}

usepackage{tikz}
    usetikzlibrary{arrows.meta}

begin{document}
    
    begin{tikzpicture}
    
        draw[thick] (-2.5, -{sqrt(3)/2}) rectangle (2.5, {sqrt(3)});
        
        path[clip] (-2.5, -{sqrt(3)/2}) rectangle (2.5, {sqrt(3)});
        
        foreach x in {-1,0,1}{
            foreach y in {-3.5,-2.5,...,2.5}{
                draw ({y - 0.5*Mod(x+1,2)},{x*sqrt(3)/2}) -- ++ (60:1) -- ++ (-60:1) -- cycle;
            }
        }

        coordinate (next) at (-2, 0);
    
        foreach x in {1,2,...,4}{
            draw[-Latex,blue] (next) -- ++ (60:1) coordinate (next);
            draw[-Latex,blue] (next) -- ++ (-60:1) coordinate (next);
        }
    
    end{tikzpicture}
    
end{document}

which yields:

To build the arrows, I created a new coordinate system in which the x-axis is horizontal and the y-axis is the side of the 60° angle. It is called triangular and its alias is tri.

Update New coordinate system

documentclass[tikz,border=5mm]{standalone}
makeatletter
define@key{triangularkeys}{x}{defmyx{#1}}
define@key{triangularkeys}{y}{defmyy{#1}}
tikzdeclarecoordinatesystem{triangular}%
{%
setkeys{triangularkeys}{#1}%
pgfpointadd{pgfpointpolarxy{60}{myx}}{pgfpointpolarxy{120}{myy}}
}
makeatother
tikzaliascoordinatesystem{tri}{triangular}

usetikzlibrary{arrows.meta}
begin{document}

begin{tikzpicture}[x=0.75cm, y=0.75cm,>=Latex]

% the grid
foreach j in {0,...,1} {
  foreach i in {0,...,5} {
    draw[gray](0:i)++(60:j)++(120:j)--++(60:2)--++(-1,0)--++(-60:2);
    
    draw[gray](0:i)++(90:{(1+2*j)*sin(60)})--++(1,0);
}}
draw(current bounding box.north west)rectangle(current bounding box.south east);


% the arrows
draw [thick,red,->](tri cs:x=1,y=0)--node[below]{x}++(tri cs:x=1,y=0);
draw [thick,blue,->](tri cs:x=1,y=0)--node[below]{y}++(tri cs:x=0,y=1);

draw [thick,violet,->] (1,0)++(tri cs:x=1,y=0)--++(tri cs:x=1,y=0);
draw [thick,violet,->] (tri cs:x=2,y=0)++(1,0)--++(tri cs:x=0,y=-1);
draw [thick,violet,->] (2,0)++(tri cs:x=1,y=0)--++(tri cs:x=1,y=0);
draw [thick,violet,->] (2,0)++(tri cs:x=2,y=0)--++(tri cs:x=0,y=-1);
end{tikzpicture}

end{document}

documentclass[tikz,border=5mm]{standalone}
makeatletter
define@key{triangularkeys}{x}{defmyx{#1}}
define@key{triangularkeys}{y}{defmyy{#1}}
tikzdeclarecoordinatesystem{triangular}%
{%
setkeys{triangularkeys}{#1}%
pgfpointadd{pgfpointxy{myx}{0}}{pgfpointpolarxy{60}{myy}}
}
makeatother
tikzaliascoordinatesystem{tri}{triangular}

usetikzlibrary{arrows.meta}
begin{document}

begin{tikzpicture}[x=0.75cm, y=0.75cm,>=Latex]

% the grid
foreach j in {0,...,1} {
  foreach i in {0,...,5} {
    draw[gray](0:i)++(60:j)++(120:j)--++(60:2)--++(-1,0)--++(-60:2);
    
    draw[gray](0:i)++(90:{(1+2*j)*sin(60)})--++(1,0);
}}
draw(current bounding box.north west)rectangle(current bounding box.south east);


% the arrows
draw [thick,blue,->](tri cs:x=0,y=0)--node[below]{x}(tri cs:x=3,y=0);
draw [thick,blue,->](tri cs:x=0,y=0)--node[left]{y}(tri cs:x=0,y=3);

draw [thick,violet,->] (tri cs:x=1,y=1)--(tri cs:x=1,y=2);
draw [thick,violet,->] (tri cs:x=1,y=2)--(tri cs:x=2,y=1);
draw [thick,violet,->] (tri cs:x=2,y=1)--(tri cs:x=2,y=2);
draw [thick,violet,->] (tri cs:x=2,y=2)--(tri cs:x=3,y=1);
end{tikzpicture}

end{document}