An effective warmup is one of the most important parts of a trumpet or brass instrument practice routine. There are a few simple principles to keep in mind as you consider how to warm up.
1) Consistency - Your warmup routine should not vary tremendously from day to day. That means it should be short (5-10 minutes) so that you can complete it even under less-than-ideal circumstances.
I have used the same basic warmup routine for 10+ years. Once I begin, my body automatically knows exactly what we're doing, so I'm already nearly 100% warmed up after the first exercise.
2) Technique-oriented - The warmup is primarily about getting your embouchure to begin to work together. That means the focus should be on the lips, the air, and the tongue, not on the fingers or on learning new music.
3) Three fundamental exercises - In my experience, an effective warmup hits three fundamental areas: long tones, lip slurs, and tonguing.
Long tones focus on the consistent airflow, the breath.
Lip slurs, being a bit of a misnomer, focus primarily on the tongue position, but also on keeping the lips in a consistent position throughout the range of the instrument so that you don't have to "reset" your embouchure when going from low to high or vice-versa.
Tonguing, of course, focuses primarily on the tongue, and is probably the most important area to address. If I don't have time for a full warmup, 30 seconds or a minute of single tonguing repeatedly on a middle G gets me mostly ready to play.
I'll address these three fundamental exercises in order
Long tones - Start in the middle range of the horn. I like to start on middle G or middle C and work my way out, though very early beginners may need to start on low C. I use the Caruso Six-Note Exercise, which I'll explain in a future blog post.
Dynamics should be not too loud, not too soft. Focus on maintaining a consistent sound throughout the duration of the note. You may want to use a tuner, but be wary of the trap of tuning with your eyes rather than your ears.
Lip Slurs - As I mentioned before, the term "lip slur" is a bit of a misnomer because you actually change notes by arching the tongue up or down, thus changing the air speed. I'll elaborate more on the role of the tongue in the embouchure in a later blog post.
Focus on doing these very cleanly and not too fast. A very common mistake I hear from beginning or intermediate players is doing their lip slurs very quickly and with lots of "fracked" notes and "splee-ahs". Increase the tempo only when it can be performed cleanly.
Tonguing - There are many ways to address tonguing, but the best starting point is to keep it simple. My "desert island" embouchure exercise is what I call the "repetitive tonguing exercise." You simply choose one note and single tongue 16th notes on that note for 30 seconds to a minute. Then rest an equal amount of time and repeat a half step up. I usually start on G and repeat the pattern until I get to middle C.
A few points to keep in mind when performing this exercise:
1) You may want to use a metronome. Depending upon your experience level, most people start between 60-85 beats per minute.
2) Accent the first note of each 16th note group. TA-ta-ta-ta TA-ta-ta-ta TA-ta-ta-ta TA-ta-ta-ta, etc.
3) This is almost like a tongued long tone. Make sure the airflow is consistent, but the notes are crisp and nearly staccato. When properly performed, it sounds almost like a snare drum exercise.
As always let me know if you have any questions. Best of luck in developing your own effective trumpet warmup routine!