Warm-up is an important step in lengthening the amount of playing time as well as getting a nice full tone on the first note.
I usually start my warm-up by controlling by breath through practicing my air flow through the horn. I breath for about 4 counts and then breath out for about 16 counts. As I get further and further into the breathing, I will breath in for fewer counts and I will exhale for more counts. Breathing provides a controlled air supply to support the notes. Also, for more of a challenge, try blowing with all the valves half-way. This provides some extra pressure against the air flow.
Buzzing is basically practicing the sound without the horn. Try to play some steady slurs and scales on the mouthpiece. Start on one note and gradually down and up repeatedly, opening the gap more and more. Remember not to move the sides of the mouth. The mouthpiece requires a more focused embouchure because there are no valves to help pick a note.
After slurs, I usually do scales. Scales really enhance the focused embouchure when reaching octaves. Try to keep the notes long and sustained with minimal break in between. There is no horn to aid the connection so the mouthpiece is very hard to play on. Playing anything on a mouthpiece, as well as warming up on a mouthpiece, makes it much easier when it translates to the horn.
Finally, I recommend the B.E.R.P., which is a device that can be attached to the lead pipe of the horn. A mouthpiece can be placed in the B.E.R.P. for buzzing. The device is very efficient because it requires buzzing while the horn is being held up. This makes sure that there is even pressure on both lips. When people usually buzz without the horn, they might bend the mouthpiece in a way that would not be when they are playing the horn. Finally, there are holes on the device that can be covered to provide pressure.
Scales and their patterns make it easier to play music. Most music have a pattern or scales within the context. Not only does this part of the warm-up practice creating a fluid sound across the octave, but also practices fingerings. It is important to push down the valves with a quick speed, despite the tempo of the music.
Practice all scales and learn new ones. It will make it easier to recognize them in the music and they will become natural to play. A lot of notes in music is not so hard when you know the pattern and the fingerings are memorized.
Warm-up should not just be a routine. Your mouth will not feel the same everyday. There will be good and bad days, when you can hit a lot of high notes and when you can't even find the note you want. Warm-up is what makes this aspect a little more consistent. A focused and concentrated warm-up will allow for you to play as well as possible.