1. Intonation
It's essential to teach beginner students basic slide positions (1-7) and where they are on the horn. Over time, students should begin to understand that slide positions are not concrete placements--like frets on a guitar or keys on a piano--but rather should be treated as fluid areas that can be adjusted based on what they hear, and the context of a given note or passage. Accurate tuning slide settings also become more critical for developing players at this stage. Encourage the student to use their ears as well as a tuner to help them figure out how to adjust the slide for intonation. They should move their slide, as opposed to bending the pitch with their lips.
2. Slide technique
Good slide technique allows the player to be dexterous on their instrument. It is vital that we not only teach the student how to hold the slide, but how our arms ergonomically work to move the slide fluidly and efficiently.
3. Embouchure setup
This varies from player to player, but I generally teach my students that the outside of their embouchure (corners, above their top lip, and chin) should facilitate security, while the inner part (aperture) should be soft and malleable to free the lips up for vibration.
Regarding the embouchure, make sure that the student understands the "corners" should serve to provide stability, much like an athlete using their core to stabilize their athletic movements. I like encouraging security instead of tension. On the other hand, the lips should be soft and gentle, allowing the air to pass through them.
4. Posture
Good posture plays an important role, as it allows the player to open up their lungs. Make sure students aren't slouching into their chair, as this makes the lungs collapse. A player shouldn't over-arch their back either. Most people overlook the angle of the player's head/neck. Rather than going to the instrument, communicate with the student that they should bring the instrument to them and not force it into their lips but gently place it on their embouchure, just enough to create a seal with the lips.
5. Sound concept/vowel shape
One of the cornerstones of brass playing is sound concept. The sound that comes out the instrument should be a direct reflection of the idea of how we want to sound in our head. Listening to the best players in the world on our instrument as well as other artists can give us a larger color palette and inspiration to experiment with our sound.
For trombonists, vowel shape can have a direct impact on sound quality. I typically teach "Toh" or "Doh" for low register, "Tah" or "Dah" for mid register, and "Tee" or "Dee" for upper register.
6. Articulation (legato)
Legato tonguing requires good slide timing with the rhythm of the piece. It is important to remember that on trombone, notes on the same partial (or floor, as I explain them) must be articulated no matter what, but with a softer "Th" or "D". Moving the slide in rhythm will prevent the student from smearing or glissing the notes. When crossing partials, encourage the student to take advantage of natural slurs to make the phrases as smooth as possible.
7. Air flow
Utilizing Breathing Gym and various tools such as a spirometer, or even just breathing tubes in the classroom will place a priority on air, the primary mover of sound for wind instruments. Also, if the player has issues with the beginning of their notes it's almost always air efficiency--though sometimes it can be tongue placement.
8. Efficient daily routine
There are many resources for directors and young students to develop a healthy routine. I would strongly encourage directors to guide their students toward a simple but effective daily routine that the student can commit to for 15-20 minutes. It's not necessarily the AMOUNT of time students spend practicing their instrument, but rather the QUALITY of time that is most important. Young students should be encouraged to focus primarily on tone, while addressing articulation, flexibility, dynamics, etc. These elements are all beyond a warmup routine in my pedagogical approach. The warmup routine should be treated as a mission to get the player's mind and body ready for daily playing.
9. Tongue-slide coordination
Understanding that the articulation and slide have to arrive at the same time can free up the student for expressiveness in their music. A telltale sign of this issue is audible slide noise in their sound (unless they're not articulating at all). It's either one of two problems: the slide arrives to the destination before the tongue or vice versa. To fix this, have the student play long tones with the focus of arriving as late as possible with the slide and releasing the air with their tongue at the precise moment the slide arrives. Practicing glissandos may also help with coordinating air and efficient slide movement, as the student need only focus on timing the articulation with their arm.
10. Playing on left side of stand
When I hear students play, I strongly encourage them to play to the left of their stand so their sound will carry. Playing with the bell into the stand has a muting effect and won't allow the player to sound as full. Often this small fix can make a huge difference in the sound of your trombone section.
Additional Links
In addition to the information above, below are some videos by James Markey, bass trombonist of the Boston Symphony Orchestra. I regularly show my beginners these videos to help them resolve some of the problems I listed above: