J.B. Arban's Complete Conservatory Method (c. 1859) remains one of the most influential works of brass pedagogy ever written. It constitutes a remarkably complete representation of the skills required for cornet performance during Arban's time, and, with the addition of modern supplemental materials, still provides the best framework for addressing the physical skills needed by today's trumpet player.
Page numbers (below) are from Arban's Complete Conservatory Method for Trumpet (Cornet), edited by Edwin Franko Goldman and Walter M. Smith, with annotations by Claude Gordon, published by Carl Fischer.
pp. 14-16: These scale-derived exercises are excellent for articulation/airstream practice. I suggest eliminating the staccato markings (here, and, with few exceptions, throughout the book) and concentrate on blowing an uninterrupted airstream.
pp. 20-21, #46: Also provides good articulation practice, as well as interval studies; Can you get all-the-way through this one (see the key change with each line) without making any mistakes? It takes a great deal of concentration.
pp. 59-86. Exercises on major, minor and chromatic scales. Do these after you know all of your major and minor scales (which should be as soon as possible!).
pp. 91-93: "Preparatory Exercises on the Grupetto." Arban wrote his Method (c. 1859) to provide study on virtually every technique required of cornetists at the time. The grupetto is a type of ornament, but these exercises are good not only for becoming familiar with that ornament, but also for finger dexterity and interval playing.
pp. 125-130: Interval Studies. Concentrate first on p. 125 before attempting the succeeding pages.
pp. 142-43: Major and minor triads in all inversions in all keys. Good for finger/interval technique, and learning to "get around on the horn."
p. 147: Dominant Seventh exercises. Each line spells out the tonic (I) chord in the first and last measures, and presents all inversions of the dominant seventh chord (V-7).
p. 149: Same as above, but with diminished seventh chords. These chords can be particularly difficult, finger-wise - dim. sevenths are simply stacks of minor thirds.
Cpp. 142-43, 147, and 149 are excellent exercises for helping you develop a working technical "vocabulary." Just as with scales, you'll run into these patterns over and over again, so these exercises are a good opportunity to learn them once and for all.
pp. 175-77: Double-tonguing exercises. Instead of the "tu-ku" syllables suggested by Arban, I find it much easier to use the syllables "dah-gah" - the G is easier to say repeatedly when playing than is the K the G - seems to be "farther forward in the mouth"). Practice double-tonguing slowly as well as quickly in order to be able to listen to your articulated sounds more carefully. You'll note that this section gets fairly difficult quite quickly, so if you're doing these for the first time, you'll need to supplement these studies. One thing you might do is to practice the triple-tongue section (pp. 155-59), and substitute a group of two (dah-gah) for the three printed on the page (dah-dah-gah). You might also want to practice your scales with a variety of tonguing patterns on each pitch (one eighth, two sixteenths, etc.)
pp. 155-59: As with double-tonguing, forget about "tu" and "ku"; say "dah-dah-gah" and I think you'll find this to be a lot easier. Again, remember to practice these slowly as well as quickly - concentrate on evenness in the groups of three.
pp. 285-289: "Fourteen Characteristic Studies." These are "wall-to-wall difficult," and are a good opportunity to try to apply the techniques taught in the rest of the book.
pp. 301-347: "Twelve Celebrated Fantaisies and Airs Varies by Arban." Includes Carnival of Venice and other well-known favorites. Try out your multiple-tonguing on these!