How to draw a line, a curved line and an arc on a pgfplot?

This is what I would do --- then you can add things and change the coordinates (you will need to calculate where do you want your lines to go, exactly). I heavily commented the code to give you a lot of "fish lines" to catch your own fish.

documentclass[12pt,a4paper]{article}
usepackage[T1]{fontenc}
usepackage{amsmath}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{pgfplots}pgfplotsset{compat=1.18}

begin{document}
begin{tikzpicture}
begin{axis}
[
    % explicitly specify axis limits
    xmin=0,xmax=1,ymin=0,ymax=1,zmin=18,zmax=26,
    xlabel={x},
    ylabel={y},
    grid,
    % do not clip the draw commands and clip each
    % addplot to its domain
    clip mode=individual,
]
% line on the xy plane (that way goes behind the surface)
% we mark a point at 30% of the line
draw[thick]  (0.4,0,18) -- (0.4,1,18) coordinate[pos=0.3](a);
% main surface
addplot3[blue, no marks, surf, domain=0:1, samples=10] {25-5*x*x -y*y};
% graph restricted at at y=0.3 (parametric curve format)
addplot3[ultra thick, no marks, domain=0:1, samples=10, samples y=1] (x, 0.3, 25-5*x*x -0.3*0.3);
% line  touching the parabola
draw[thick]  (0.4,0,18) -- (0.4,0.3,25-5*0.4*0.4-0.3*0.3) coordinate[pos=0.3](b);% coordinate on the black parabola
% draw spline and angle label
% in and out are the angle at which the curve leaves the final and initial point
draw[thick,->] (a) to[out=100, in=-10]  node[midway, above right]{$alpha$} (b);

end{axis}
end{tikzpicture}
end{document}

You maybe prefer to have the surface semi-transparent, to better convey the meaning of the arc:

documentclass[12pt,a4paper]{article}
usepackage[T1]{fontenc}
usepackage{amsmath}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{pgfplots}pgfplotsset{compat=1.18}

begin{document}
begin{tikzpicture}
begin{axis}
[
    % explicitly specify axis limits
    xmin=0,xmax=1,ymin=0,ymax=1,zmin=18,zmax=26,
    xlabel={x},
    ylabel={y},
    grid,
    % do not clip the draw commands and clip each
    % addplot to its domain
    clip mode=individual,
]
% line on the xy plane (that way goes behind the surface)
% we mark a point at 30% of the line
draw[thick]  (0.4,0,18) -- (0.4,1,18) coordinate[pos=0.3](a);
% line  touching the parabola
draw[thick]  (0.4,0,18) -- (0.4,0.3,25-5*0.4*0.4-0.3*0.3) coordinate[pos=0.3](b);% coordinate on the black parabola
% main surface, semi-transparent
addplot3[blue, no marks, surf, domain=0:1, samples=10, opacity=0.7] {25-5*x*x -y*y};
% graph restricted at at y=0.3 (parametric curve format)
addplot3[ultra thick, no marks, domain=0:1, samples=10, samples y=1] (x, 0.3, 25-5*x*x -0.3*0.3);
% draw spline and angle label
% in and out are the angle at which the curve leaves the final and initial point
draw[thick,->] (a) to[out=100, in=-10]  node[midway, above right]{$alpha$} (b);

end{axis}
end{tikzpicture}
end{document}