EDIT: Based on the comment (and the update) it seems like pinching is much more likely than backward normals. Oddly shaped polygons can be difficult to render in OpenGL, but they often look okay when fully rendered (per-pixel shading, versus vertex shading).
The artifacts are actually unrelated to the question about holes in faces. The artifacts are a consequence of surface normals that aren't all facing the same direction. In the 3D viewport, tab into edit mode and press n to open the settings panel. There's a section in that panel marked "View," where you can turn on "Draw Normals" (there are two buttons, one for vertex and one for face, you want face normals). This will make blue lines that show the direction each face normal is pointing. They need to all point the same direction. If you find a normal pointing the wrong direction, select that face (face select mode is easiest), press ctrl-f, and select "Flip Normals." Repeat this process with each incorrect face until all the faces have normals that are pointing the same direction.
Incidentally, if you have a fully enclosed mesh, then in edit mode you can select Mesh->Normals->Recalculate Outside to solve this problem all at once. But the mesh you've displayed doesn't look like it's enclosed. (A sphere or a cube would be fully enclosed, and a plane is not. If there were a hole in the sphere or cube, it would not be fully enclosed, and it would be difficult for Blender to be certain which side was "outside" and which side was "inside").
As for the "best" method for putting holes in a plane, that really depends on what you're doing. The method you're using here is a common one, and if the artifacts are the only problem, then it is probably "best."