Firstly, find the embouchure that works for you. Here's a great resource:
http://www.trumpetthink.com/embouch.htm
Secondly, position of the tongue is important, since it directs where and how the air should go (for the high/low register).
Thirdly, imagine your lungs as a big gas tank. You would want to fill up that gas tank full, so why not your lungs? An ideal breathing system would be this: sit straight (with or without trumpet), and when you begin to intake air, think of it like your yawning. Same idea here. So when you breath in, suck in your stomach as much as possible while keeping your chest out AT ALL TIMES (like a pro body builder). In addition, make an "oh" sound when you breath in (would make an "O" shape with your mouth and your tongue low). This open's your throat, thus intaking air more effectively and providing a big-enough passage for when you play. When you exhale, keep the stomach sucked inward, chest out, and throat open. This will not only provide a steady air stream, but a clear resonant tone worthy of making money. You will be able to hit any register as well (assuming you have an ideal embouchure). Common problems: intaking air but not sucking in stomach; Intaking air but not opening the throat (no "O" sound); intaking air for too short time. Common problems when exhaling: not placing tongue in proper position; teeth closed (choppy air stream. Think 'growl'). The first problem you would arrive at is to create tension the chest (when somebody punches you in the gut and you would get that feeling in your chest), which would close up the throat, thus not providing enough air to create a clear, resonant tone.
Hope this helps!
Best regards, belltrumpetplayer19
