Below are some of my preferred method books for the trumpet:
The Arban Book-
The trumpet player’s bible, it is truly necessary for any developing player. It covers all ground: flexibility, articulation, rhythm, intervals. Exercises introduce the player to every common technique on the trumpet (multiple tonguing, trills, mordents, etc.)
As an advanced player, I still use the Arban book to supplement my daily workout. Phrasing studies and duets make great work for sight-reading. The Characteristic Studies and Compositions have become audition standards (hello . . . Carnival of Venice anyone).
The Art of Jazz Trumpet by John McNeil-
I discovered this book a few years back. For those of you who don’t know, John McNeil is the professor of Jazz Trumpet at the New England Conservatory. In its introduction alone, this book manages to give one of the most complete histories of the Jazz Trumpet I have ever read.
McNeil’s best contribution is articulation, a topic all too frequently forgotten by young improvisers. The exercises progressively introduce more patterns (mostly combinations of twos and threes), freeing up the player’s phrasing while keeping their thinking rooted in good articulation.
Carmine Caruso Musical Calisthenics /
Flexus by Laurie Frink & John McNeil
These two books go together.
After attending a masterclass where Dave Douglas raved about Carmine Caruso (and Laurie Frink), I began researching Caruso and stumbled upon these two methods.
I have long practiced the principles of the Caruso method (one embouchure, coordinate all motions with foot tapping) since long before I bean playing these exercises. It was through these daily exercises that I have begun to discover my own approach to the trumpet.
Theo Charlier: Etudes Transcendantes
This may be the greatest classical etude book I’ve ever worked with. Each composition manages to be both challenging, and musically fulfilling. The ITG has a both translation available and a Second Trumpet Part for Etude 2: Du Style (from the Charlier Companion by James Olcott).