Struggling college student need help with multivariable calculus lol

Sorry for the huge delay. I will try to help you as I could.
(\nabla F)(x_0, y_0, z_0) represents the direction of greatest change in F at the point (x_0, y_0, z_0). And it is perpendicular to the surface F(x,y,z)=c as any directional derivative within the surface is 0. In other words, at (x_0, y_0, z_0) any direction \hat n within the surface satisfies \hat n \cdot(\nabla F)(x_0, y_0, z_0) = 0.

Same reasoning applies to G as well. Then, (\nabla F)(x_0, y_0, z_0) \times (\nabla G)(x_0, y_0, z_0) = 0 is essentially saying the (\nabla F)(x_0, y_0, z_0) is parallel to (\nabla G)(x_0, y_0, z_0). However, each of them is perpendicular to the surface at the point. Therefore, unless F=G is impossible, F and G are tangent to each other.

The key thing is (\nabla F)(x_0, y_0, z_0) is perpendicular to surface F(x,y,z) = c at (x_0, y_0, z_0).

However, I would like to give a more rigorous definition of what is tangent though I cannot think of one now.

I would also like to add we welcome your questions and will try to give your answers sooner in the future.