The trill or "shake" is initially performed between the note written and the note above it according to the key signature, subject to modifications by previous accidentals.
E.g. in E flat major a trill written over D will be between D and E flat, as your band teacher says, starting on D.
In some music it is conventional to start on the upper note, so in the example you'd start on E flat and alternate with D.
It's important, however, to consider how you are going to finish your trill. The convention (not always observed) is to combine the trill with a turn.
To develop your example, let's say you have that D with the "tr~~~" marking, in E flat major, and that the next written note is E flat. You'd then rapidly alternate between D and E flat as usual, starting on D (or starting on E flat if you prefer / if it suits the style), and then finish with the pattern E flat - D - C - D, ending up on the next written note (which is E flat in this example).
Sometimes it's not desirable to include the turn, particularly if the trill symbol has been used as a shorthand for a tremolando between two notes a step apart, as is often the case in modern arrangements. The inclusion of the turn is particularly desirable when the "trill" note is the leading note the prevailing scale and the note immediately after the trill is the tonic (as in our example).
Starting on the upper note: I often like to do that especially in Bach and often Mozart, Beethoven, and later composers. Mainly it's a matter of considering the context and understanding what the musical purpose of the ornament is within the context, and what will best achieve that musical purpose.