I would work on buzzing your lips on the mouthpiece alone and different pitches. Once you can buzz songs on the mouthpiece, (start simple with hot cross buns, etc.), you can then certainly play them on trumpet as long as you know the right fingerings. A book that helped me a lot when I was learning how to play was "All About Trumpet", by Jonathan Harnum. It's great if you are trying to teach yourself.
Here is how to buzz your lips;
1. Look in the mirror and smile at yourself.
2. Take a deep breath.
3. Place your lips together as though saying "mmm". Keep the corners of your mouth tight, like you just had a big bite of lemon.
4. Force the air in your lungs through your lips. Use your stomach muscles to help push the air out.
5. Hold the sound of the buzz steady for as long as you can. Strive for a big, fat, steady sound.
6. Repeat for a few minutes. Don't forget to breathe. ;-)
Try some simple songs using this technique, and you sound will improve in no time. Try buzzing high to low and low to high; this will help you hit notes of the same fingering (called a lip slur.) You do this by speeding up the air as your buzz to go higher, and slowing it down to go lower.
Good luck.