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I have a plane $ax+by+cz+d=0$, what is an orthonormal basis for it?

Is it a set of three 3D vectors $e_1, e_2, e_3$ with the following properties?

1. each $e_i$ has unit length;


2. 
   $e_1perp e_2$, $e_2perp e_3$, $e_3perp e_1$;


3. 
   $e_1$ and $e_2$ lie on the plane and so $e_3$ is parallel to $(a,b,c)$.

A plane is a two dimensional object, so the basis will have only two elements (two 3D vectors). In your notation $e_3$ is not part of the basis. Everything else is fine.

You can state all of the vectors in a plane by just two none-zeeo vectors in different directions. The set of these two vectors is called a basis for plane vector space. As you see, there are infinite bases for a plane!
Orthogonality and orthonormality can be difined only after defining norms. Otherwise each basis see itself orthonormal, because in it's point of view the base vectors in it are normal (unit lenght) and orthogonal to each other!
Maybe it's some strange but for understanding that you can think about seeing object from a magnifying-glass!