Summarize this article:

Hold the small mouthpiece to the middle of your lips. Now, press your lips firmly against it, inhale deeply, and blow out through it. Keep the corners of your lips tight and leave the center loose. Blow hard enough that you feel your lips vibrate and hear a blowing sound.  Make a small hole in the center of your lips as you blow air. Don't puff your cheeks out when playing—this will make you short of breath quickly, and your note will sound rough and spluttered. High pitches are created by faster lip vibrations, which are created by tightening your lips as you blow. To tighten your lips, pull back the corners of your mouth to make them more firm as you play higher. Your jaw should raise naturally and the air you're blowing should move in a more downward motion. The final sound should be a "tee."  Make sure that you stay relaxed when playing higher notes and blow faster air through the instrument. Don't make any weird faces that would cause tension in the sound. It will feel natural to press against the mouthpiece for higher notes—resist this temptation, as it will lead to problems later. Low pitched notes are created by slower lip vibrations, which are accomplished by loosening the center of your lips and dropping your jaw. However, the air still needs to remain fairly fast and steady. To maintain a good quality sound, you'll need to exhale more air than you do with higher pitched notes. Keep in mind that trombones—especially bass trombones or tenor trombones with trigger attachments—need a lot more air to play a well-supported note because of all the extra tubing. When you first start playing the lower register, it may be very challenging. To do well, increasing your lung capacity is the key. Drop your jaw and part your lips slightly to achieve a rounder, brassier sound. Always remember that playing "openly" does not mean loosening your lips. The first position is when the slide is almost all the way in. The second position is a little more than halfway between 1st and 3rd. Again, this position will vary, but the higher the note, the higher the slide tends to need to be. The third position is slightly after the bell, but will vary depending on the register in which the note is played. Make sure you don't slam the slide in when returning to the first position. The fourth position is located just past the bell, and the fifth position is located a little past the fourth position. The sixth position is almost all the way out—near the seventh position—and the seventh position is all the way out  On most trombones, there is a lip or marking where the seventh position is. The positions may vary slightly depending on the type of trombone you play.

Summary:
Blow through the mouthpiece using an "o" shape with your lips. Play higher pitched "tee" notes by tightening your lips. Play lower pitched "tay" notes by loosening your lips. Blow harder and more "openly" to play louder. Learn the first 3 positions on the slide. Practice the remaining four positions on the slide.