One of the aspects of trumpet playing that differs greatly between average trumpet players and great trumpet players is their effective use of alternate fingerings.  Most players realize the existence of these fingerings but don’t have enough knowledge to use them well.  I found this chart to be a great jumping off point for amateur players to explore.  The fingerings are listed in order of: normal fingering, then the alternates, in order of intonation.

As with anything, a thorough understanding of the science behind this subject is beneficial, i.e. the production of sound, overtone series, partials, etc.