No matter what instrument you play, when you are sitting you should be sitting towards the edge of your chair, feet flat on the floor, and sitting up nice and straight, but not tense. All of this applies if you are standing, except for sitting on the edge of your chair. You can pretend that you are balancing a book on your head, or perhaps that you are a puppet with a string attached between your head and the ceiling. Another way to check your posture is to stand up nice and straight, then sit down carefully on the edge of your chair without changing the upper half of your body.
For each instrument there are some specific things to be aware of:
Flutes
Your fingers should always be curved, not flat, when on or over the keys. Your wrists should be fairly straight, not bending in or out at a noticeable angle. Elbows should be out from the body, but not so far up that they are nearly parallel to the ground. The flute should be almost parallel to the ground, with the right end tipping down slightly towards the ground.
Clarinets and Saxophones
Fingers should be curved over the keys, never flat. Elbows should be several inches out from the side of your body. When playing, make sure that you are covering the holes with the tips/pads of your fingers (clarinets only need to worry about covering the holes; both clarinet and saxophone need to play with the pads of their fingers). For keys that are not pushed down for a particular note, your fingers should be hovering just above the keys. That way, if you need to quickly put those fingers down for an upcoming note they are already very close to the keys (fingers should still be curved). The clarinet should come out of your mouth at about a 45 degree angle.
Trumpet
With the mouthpiece end of the trumpet facing you, your left hand should go in the middle of the trumpet around the valve casings. If there is a ring in the middle, your left hand ring finger goes through this ring, and then the pointer and middle fingers rest against the far side of the valve casings. The thumb rests on the side closer to you.
Your right hand is the one that presses the valve. Your ring, middle, and pointer fingers should be CURVED over the valves, so that you play on the pads of your fingers. Your fingers should never be flat, and you should never play on your knuckles; always play on the pads of your fingers. Your pinky finger rests on top of the metal loop on the far side of the valves, not in the loop. Your thumb rests between the first two valve casings under the extension of the tubing where your mouthpiece goes in. When playing, the horn should be parallel to the ground.
Trombone
Start by making your left hand into a shape as if you were holding a squirt gun: your pointer finger should be out, your ring, middle, and pinky fingers should be curved in, and your thumb should stick straight up. Your three curved fingers will curve around the left side of the slide right between the two bars on the slide. Your pointer finger will go straight across the top bar towards the mouthpiece, and your thumb will curve over the bar on the bell.
Your right hand will be like the "spock" hand; you should try to separate your pinky and ring fingers from your middle and pointer finger, and your thumb will hang out on its own for now. Essentially, there should be a V between your middle and ring fingers. Under the left part of your slide, where your left fingers are curved around it, the part of the slide right under the bottom bar will go in the middle of the V. With the pads of your pointer and middle finger and your thumb, pinch the bottom bar. Your pinky and ring finger should be resting on the outside part of the slide. When playing, the horn should be parallel to the ground.
Baritone
The baritone should be placed in your lap/on your thigh so that the mouthpiece meets your mouth without you having to bend forward or lean back to reach it. Your right hand pointer, middle, and ring fingers will depress the valves and your left hand will support the instrument around the left.
French Horn
Your left hand should curve around the side of the instrument so that the pads of your left hand pointer, middle, and ring fingers rest on the valves. Your pinky will rest in the loop on the far side of the valves.
Your right hand goes right into the bell of the instrument. It should fit deep enough inside that it is snug but not painful, and the back of your hand will rest against the "2 o'clock" position or so, curved in a shape as if you're holding a mug. The bell of the instrument will rest on your right thigh, and you should be able to bring the mouthpiece right to your mouth. If necessary, you may need to brace your right foot against your chair leg to raise your leg a little.
Percussion
The drum should be set right around where your belt would be on your pants. Elbows should be at about a 45 degree angle or so, out enough so that your sticks make a 90 degree angle (an "L") where they meet at the center of the drum (or key for bells). The key to percussion is to stay relaxed. When playing the bells, your sticks should be slightly offset form one another (one very slightly in front of the the other) like we talked about in class to ease the alternating sticking as you move up and down the instrument. Your hands for both snare sticks and mallets should curve around the stick so that your fingers are always touching, but loose enough that there is room for movement of the stick in your hands so that the sticks can bounce if needed. About halfway between the halfway point of the stick and the end of the stick closest to you is where your hands will go; it will vary a little based on the stick and student. The sticks will rest in the knuckles of your hands with your pointer finger and thumb will wrap around it, as well. Hands will lay flat, with the backs facing the ceiling.
ALL MOVEMENT WILL BE FROM YOUR WRISTS, NOT YOUR ELBOWS.