How can you tell if two atoms are in the same plane?

Simply choose three or more atoms to be on the same plane (at least three atoms are required to make a 2D angle), and make the remaining ones not in that plane.

The ones not in the plane are drawn with wedges (front) and dashes (rear).

The simplest 3D example is methane, tetrahedral:

In this case, the lefthand #"H"-"C"-"H"# is in the same plane, while the bottom-right #"H"# is in front, and the middle-right #"H"# is in the rear.

This first way is most common, but if this helps, methane can also be drawn as:

...if you rotate the first image by #120^@# counterclockwise. I drew it both ways since tetrahedral in particular can be difficult to visualize and rotate in your head.

A more complicated example:

This is #"PCl"_5#, phosphorus pentachloride, trigonal bipyramidal.

Here, you can see that the right-hand #"Cl"# (one equatorial and two axial) are in the same plane, since they are all lines. The bottom-left #"Cl"# is in front, and the upper-left #"Cl"# is in the rear.

Basically, if you are unsure, consider buying an actual model kit or borrowing one, and making the structures yourself so you can see it.