Establish a good daily warm-up and routine if you don't already have one. If you need one, I have a warm-up packet that I'll send to you. Email me at: email@example.com and ask for it. Every trombonist has a slightly different way of going about their routine, but most routines have these commonalities: long tones, lip slurs, and tonguing.
Long tones: play relaxed and non-metered long tones every day. Focus on your sound. Have a sound in mind that you want to emulate (from your teacher, a recording, a trombone performance you attended, etc...), and chase that sound, every day. I like to think of long tones as the most relaxing thing I do, all day, and I believe that your sound is the most important thing about you, as a musician. Without a beautiful core sound, it is difficult to do much else. I like to play my Remington series with a floated first note (only air to start the first note, no tongue), and a relaxed glissando to the second note (then, breathe before the next note). I also play slow scales, using as little tongue as possible, while playing legato or glissed.
Lip slurs: play these to develop flexibility on your instrument, learning to navigate the harmonic series, without using your tongue. Start with easy slurs (4th-line F down to low Bb, and then back up), and work your way up to adding more partials and playing faster (while staying relaxed). Always play these slowly to start. A relaxed accuracy is key, in the beginning, but even after you feel you have more control, still keep things as tension-free as possible.
Tonguing: after you have gotten some air moving through the instrument while playing long tones and lip slurs, *then* work on tonguing, while keeping the same sound that you've been producing, previously. Don't allow the notes in this part of your routine to not sound as good as everything else. Practice tonguing repeated notes (static, or non-moving slide), and also practice tonguing moving notes (dynamic, or moving slide). Make sure to move your slide quickly between notes, avoiding any unnecessary glissandos (while at the same time making sure that your right arm stays as relaxed as possible). Scales are a great thing to practice here, varying your articulations.
Finally, when learning these etudes, take the time to practice slowly and accurately. Make sure that you are being attentive to every detail in the music, first.
Clinic Bass Trombone, Audition Solo 2D, Moderato
-Practice the dynamic, articulation, and rhythmic contrasts - the composer is very specific about these things. However, remember that dynamics are relative, and you don't have to be too loud or too soft anywhere in this solo - just make sure that your dynamics provide contrast in your performance, and always strive for a beautiful and full bass trombone sound.
-In m.5 approach the staccato articulations by making sure that you put space between your notes, keeping the articulation light, instead of thinking of these as short notes. This will ensure that you are still playing with good tone here, and not allowing the bass trombone to "bark" too much.
-Start subdividing triplets during the half note in m.8 - this will ensure that you are in time in m.9.
-Be reasonable when practicing the cresc. in both the "a" and "b" endings. The composer didn't note an end to the crescendo, so it is up to you, the performer, to determine whether to cresc. all of the way to the end, or find a good "stopping place" for this. Either way, don't overdo it.
-If you have an independent double-trigger bass trombone, explore your two options (F trigger vs. Gb trigger) - you may find that using the Gb trigger for low B's and C's will make for more fluid slide movement. Also, using the Gb trigger for the low E's in m.8, 9 and 17 will make for fluid motion to the F# in 5th position, and the rest of those phrases.
-Along these same lines, practice your G major scale, using that Gb trigger (instead of the F trigger), as a cross-training exercise.
Senior Bass Trombone, Audition Solo 3B, Andante appassionato / Allegro scherzando
-Practice the dynamic, articulation, and rhythmic contrasts - the composer is very specific about these things. However, remember that dynamics are relative, and you don't have to be too loud or too soft anywhere in this solo - just make sure that your dynamics provide contrast in your performance, and always strive for a beautiful and full bass trombone sound. Having said that, by the time you cresc. to ff at the beginning of either ending, you want to have a very full and robust sound (while still remaining in control).
-Practice playing clean descending lip slurs in the low register, specifically the G to G slur at the beginning. You want to make sure that the first thing that you play sounds really good.
-Note the 5/8, and practice playing in 5, especially if it is a new thing for you. Make up an exercise using the G harmonic minor scale, and get comfortable with "grooving" in 5.
-Practice the Allegro scherzando section using a metronome. Instead of setting your metronome to 108, use the eight note, and set it to 216. This way, you can use the metronome all of the way through the 4/4, the 5/8, and back to the 4/4 endings (since the eighth note remains constant throughout these sections).
-While on the subject of time, or pulse, use your metronome to practice coming in exactly in time on (or moving on) the second triplet note (m. 2, 4, 5, 7, 8, 29, 30, 33 and 34).
-Plan your breathing in the beginning so you can make it through the low passage in m.4-5, and also the sequence in m.23-27 (or m.23-31, when you play the "b" ending).
-If you have an independent double-trigger bass trombone, explore your two options (F trigger vs. Gb trigger) - you may find that using the Gb trigger for the low C's in m.5 and 10 will make for more fluid slide movement. Also, decide whether to play the low D's in the G minor arpeggios with the F trigger, or in 1st position with both triggers. I use both options on my recordings, depending on the phrase. For m.23-26, you may find that using both triggers for the first note, and then the Gb trigger for the second note, will make each measure easier, and then make for a smoother phrase. Whatever you choose, practice it slowly, making sure that you don't have any trigger errors in the audition.
Download these bass trombone performance notes [pdf].