Here's an exercise to get you using your air. Play a first space, treble clef, F on your horn and memorize the pitch. Now remove the tuning slide from your trumpet. Take a full, relaxed breath, as if you are yawning, then play the horn. Buzz the pitch F on the leadpipe. Think of blowing warm, moist, relaxed air through the horn (as if you are trying to fog up a mirror). Listen to the buzz and strive for a reedy, buzzy, kazoo-like sound. When you get a loud kazoo sound on the pipe, you can check how relaxed your air is by taking the mouthpiece out of the pipe and playing just the mouthpiece. Do not buzz the mouthpiece. Only blow in a relaxed manner and listen to the air whistling through it. Add the leadpipe and all of a sudden you'll get the pitch F buzzing on the pipe. For some, the pipe may buzz an E or an Eb, so if the F sounds flat, that's ok.
I've used this leadpipe exercise on many beginning students and have found that once they discover how to buzz the leadpipe they are capable of producing a full resonant tone.