As a college teacher, I am often called upon to help students preparing for all-region and all-state auditions, solo and ensemble performances, and college auditions. Students are always looking for a competitive edge over others. It is easy to get into the mindset that a new mouthpiece or instrument is what you need - I have fallen victim to this more often than I would like to admit. The single greatest competitive edge you can gain over other auditioners is time well spent in the practice room.
The total investment if you have neither of my "two things" is around $50. These books are both "war horses" of the trombone repertoire: Schlossberg and Bordogni. I order every piece of sheet music I can from www.hickeys.com. The more music we come by honestly (or in other words, pay for with $$$), the slower the prices will rise on music that is currently available.
The Schlossberg Daily Drills and Technical Studies may seem like a daunting book, but I have an easy solution: choose exercises based on the day of the month. If it's the 11th, play 1, 11, 21, 31, 41, etc. Set a metronome on 54 and go for it, making sure that you play full note values and hold yourself to the highest standards of consistency. If something doesn't sound right, go back and do it again. Rest often.
For the Bordogni Vocalises Mulcahy Complete Edition or the 2011 Revised edition of the the Rochut book: play as written, in tenor clef (up a fifth), and tenor clef down an octave. Later, add down an octave and up an octave in bass clef. Breathe where needed, remove ornaments that inhibit smooth connections between notes, and most of all, enjoy your developing sound. If a phrase doesn't work, gliss-buzz through it slowly with careful attention to the pitch and keeping the vibrations going between notes.
A friendly, fair warning: these probably won't sound great the first several days, but it's kinda like the gym - you won't see results right away, but each session brings you one step closer to the body you want.
If you have any questions, or are unsure how to proceed, I would love to hop on Skype and help you get started. Contact me if you could use some help!