How to draw a thick point?

When you want to build a figure, it is easier to give a name to the points you want to use. To do this, I used nodes, because any path from one node to another goes from edge to edge without crossing them. To do this, I defined a style called point like this point/.style={circle,fill,inner sep=1.3pt}

I used the angle library to draw the angles. And I used the calc library to build the orthogonal projections, which avoids having to do manual calculations.

 documentclass[tikz,border=5mm]{standalone}

 usepackage{amsmath}
 usetikzlibrary{arrows,angles,calc,quotes}

begin{document}
begin{tikzpicture}[
point/.style={circle,fill,inner sep=1.3pt}
]
node[point,label={[below left,yshift=-2pt]:$O$}] (O) at (0,0){}; 
node [point,label={[above right,yshift=-2pt]:$boldsymbol{M}$}](M) at (4,3){};
node [point,label={[below,yshift=-2pt]:$boldsymbol{M'}$}](M') at (2,0){};
node [point,label={[left]:$boldsymbol{M''}$}](M'') at (1.9,3){};
draw[dashed] (M'')--(M)--(M');
draw[dashed] (M)--($(O)!(M)!(M')$); % projection orthogonale de M sur (OM')
draw[dashed] ($(O)!(M)!(M'')$)--(M);% projection orthogonale de M sur (OM'')
%     draw[dashed] (4,3)--(2,0);
draw[very thick,->] (O)--(1.5,0) node[below] {$mathbf{e}_1$};
draw [very thick, ->] (O)--(0.94,1.5) node[left] {$mathbf{e}_2$};
pic [draw=violet,fill=violet!30,text=violet!60!black,angle radius=8mm,"$beta$",angle eccentricity=1.3]{angle=M'--O--M};
pic [draw=blue,fill=blue!40,text=blue,angle radius=9mm,"$alpha$",angle eccentricity=1.3]{angle=M--O--M''};
draw [very thick, red, ->] (O)--(M) node[midway,below] {$mathbf{A}$};
node[point] at (O){};% We place again the point  O     draw[->] (O) --(6,0) node [above] {$m'$};
draw[->] (O)--(3,4.8) node[above] {$m''$};
%     node at (4.2,3.1) {$M$};
%     node at (1.4,3) {$boldsymbol{M''}$};
%     node at (2.3,-0.14) {$boldsymbol{M'}$};
end{tikzpicture}
end{document}

Any angle is defined by three points which are endorsed below as coordinates

Thereafter the angles are physically drawn with the help of the pic command

Simply add all the code below to your own code to get the desired output

coordinate(m) at (4,3);
coordinate(x) at (4,0);
coordinate(o) at (0,0);
coordinate(y) at (2.5,4);

pic[ draw,->,>=stealth,blue, "$theta_1$"{fill=white},inner sep=1pt, circle,  draw,angle 
eccentricity=1.1, angle radius = 10mm] {angle = m--o--x};   

pic[ draw,->,>=stealth,blue, "$theta_2$"{fill=white},inner sep=1pt, circle,  draw,angle 
eccentricity=1, angle radius = 7mm] {angle = y--o--m};

EDIT -- I simply forgot the point nodes required by the OP-- apologies

documentclass[]{article}
usepackage{tikz}
usepackage{newtxtext,newtxmath}

usetikzlibrary{arrows, shapes, positioning, calc, decorations.text, angles, 
               quotes}
begin{document}

begin{tikzpicture}[
                    mycirc/.style={circle,
                                   fill=red!50!black, 
                                   minimum size=5pt,
                                   inner sep=0pt}
                    ]
draw[dashed] (1.9,3)--(4,3);
draw[dashed] (4,3)--(4,0);
% The nodes as points or circles are drawn as under--note tha the names of
% the nodes are given in `()` and the same can be used in place of using
% coordinate names
node[mycirc]at(0,0) (a) {};
node[mycirc]at(4,0) (b) {};
node[mycirc]at(2,0) (c) {};
node[mycirc]at(1.9,3) (d) {};
node[mycirc]at(4,3) (e) {};
node[mycirc]at(2.5,4) (f) {};

draw[dashed] (2.5,4)--(4,3);
draw[dashed] (4,3)--(2,0);
draw[very thick,->] (0,0)--(1.5,0) node[below] {$mathbf{e}_1$};
draw [very thick, ->] (0,0)--(0.94,1.5) node[left] {$mathbf{e}_2$};
draw [very thick, red, ->] (0,0)--(4,3) node[midway,below] {$mathbf{A}$};
draw[->] node[below]{$O$} (0,0) --(6,0) node [above] {$m'$};
draw[->] (0,0)--(3,4.8) node[above] {$m''$};
node at (4.2,3.1) {$M$};
node at (1.4,3) {$boldsymbol{M''}$};
node at (2.3,-0.14) {$boldsymbol{M'}$};

%Coordinate names for the angles-- can be deleted and node names can also be 
%used
coordinate(m) at (4,3);
coordinate(x) at (4,0);
coordinate(o) at (0,0);
coordinate(y) at (2.5,4);

pic[ draw,->,>=stealth,green!50!yellow!50!red, "$theta_1$"{fill=white},inner sep=1pt, circle,  draw,angle 
eccentricity=1.1, angle radius = 10mm] {angle = m--o--x};   

pic[ draw,->,>=stealth,blue, "$theta_2$"{fill=white},inner sep=1pt, circle,  draw,angle 
eccentricity=1, angle radius = 7mm] {angle = y--o--m};
end{tikzpicture}

end{document}

EDIT2 -- for inner angles instead of outer angles

Change the last lines of the code containing the pic command to--

pic[ "$beta$"{fill=white, inner sep=0pt},draw=green!50!yellow!50!red, <->, 
                >=stealth, angle eccentricity=1, angle radius = 1.5cm] 
                {angle = x--o--m};   

    pic["$alpha$"{fill=white, inner sep=0pt}, draw=orange, <->, >=stealth, 
               angle eccentricity=1, angle radius=1.5cm]
               {angle = m--o--y};

and you would get--