The first lesson a new trumpet player needs to learn, even before they play a single note on their instrument, is the proper position at which to hold the trumpet. Sit upright in a chair, insert the mouthpiece into the mouthpiece receiver at the end of the trumpet (twist it ever so slightly to seat it in place, but not too much, or too often, or it may become stuck), and grasp it between your hands. Your right hand should not be carrying the weight of the instrument. Keep it loose and as relaxed as possible. The thumb on your right hand should go just under the leadpipe (that's the tubing coming from where you insert the mouthpiece to the rest of the instrument). Your pointer, middle and ring fingers should be placed carefully on the first, second and third valves. Make sure to lay your fingers down such that you can be pressing down on the trumpet valves with a straight up-and-down, rather than a side-to-side, motion. This will allow you to build up more speed in time, and prolong the life of your instrument. Your pinky on the right hand should go on top of, not inside, the ring or pinky rest just toward the bell from where the valves are. Placing it inside this ring, though it may seem more natural, leads to too much gripping of the horn with the right hand, limiting what you can do on those all-important three valves.
When grasping a trumpet, nearly all of the weight should be carried by your left hand, as this hand is not responsible for pushing the trumpet's valves to get different notes. This is a lesson that many trumpet players, even very good ones, never quite get. They end up resting a lot of the weight of a heavy instrument on their right thumb and pinkie, which results in tension in the hand not conducive to good freedom of movement with the valves. Practice holding the trumpet with just your left hand, with your thumb in the ring closest to the trumpet mouthpiece. Now place your pointer, middle and pinkie fingers around the the valves, resting on the body of the third valve slide. This is something you need to learn in your very first trumpet lesson, or it will haunt you later. Place your pinkie in the ring closest to the bell. Make sure to adjust the ring such that your whole hand can lay comfortably. If your trumpet does not have a third valve slide ring, simply place your hand in the position that is most comfortable for you.
As well as learning to hold the trumpet properly, an important part of your first lesson is to have a solid foundation in how to "buzz" on your mouthpiece. To the uninitiated, it may seem that a trumpet player is just blowing air through his or her instrument. This is not the case. Look in the mirror, while putting the trumpet down for a minute, and give yourself a solid raspberry. Think of the sound "Pbbbfhhh." Parse your lips as if to make a "P" sound, and continue to attempt to keep your lips in that shape as you blow air out your mouth. This is the basic lesson you need to learn in order to make an effective buzz on a trumpet mouthpiece. Try to keep your lips on the same level, without letting your top or bottom lip slip over the other one.
Now, place the mouthpiece directly in the center, both up-and-down and side-to-side, of your lips. It is extremely important that you get used to placing the trumpet mouthpiece right in the center of your lips, where the airflow is the strongest. Otherwise, in subsequent trumpet lessons your range and endurance will be limited. Practice your best buzz with the mouthpiece until you can get a nice, even tone that doesn't fluctuate up and down. It will be very difficult to do this at first, but as you progress in your study of playing the trumpet, you will get to the point where you can play scales, or even entire melodies, using just the mouthpiece and without even picking up a trumpet.
You are finally near the end of your first trumpet lesson, and this is the part you have been waiting for. Place the mouthpiece in its receiver in your trumpet, press it into the instrument softly, and give it about a quarter turn. Do not develop the bad habit of constantly turning your mouthpiece in place once it is inside the trumpet. This can cause the trumpet mouthpiece to become stuck. Now, making sure to follow the guidelines in Steps 1 and 2, bring the trumpet up to shoulder level--try to keep it perpendicular to your body--and blow your very first note. Keep your fingers on your right hand lightly touching the first, second and third valves, but do not press them down. If you have a solid buzz, you will most likely produce the note "Low C." Keep with that note until you can play it without having the pitch go up and down. Now you are ready to continue further in your trumpet lessons. Having a private instructor, especially at the very beginning, is a solid investment, and one you will not regret. Playing the trumpet is a fun and rewarding pastime.