The best way to hold the slide brace is a matter of some debate among trombone players. I feel the most effective method is to put the first two fingers of the right hand below the slide brace and the thumb above it.
The third and fourth fingers sit under, not over the bottom slide tube – something like a Vulcan salute. This position gives the greatest flexibility, allowing the wrist to stay loose and controlling the slide between thumb and fingers. You want to think about leading the slide with the fingers, not pushing it with the arm and shoulder muscles.
Trombone teachers often tell their students to move the slide faster to avoid smearing between the notes in legato playing. This is good, but a problem can arise when a student tries to suddenly move the slide really fast, jerking the embouchure and messing up the legato.
"It is better to make bad music from a liberated state than to make reasonably good music from a state of bondage."
… Kenny Werner
For example, let's say you have to slur from first position F to fourth position D. Move the slide too slowly and you'll likely get a smear, but suddenly moving it at 50 miles/hour is too jerky. You can achieve a quick, yet smooth slide motion by starting out in first position at 0 mph, gradually accelerating to 50 mph about halfway between first and fourth, then decelerating back to 0 mph as you arrive in fourth. During the acceleration phase the thumb acts as a throttle, pushing the slide brace outwards–during the deceleration phase the first finger acts as a brake. If the slur went from fourth up to first you would reverse this technique, using the first finger as the throttle and the thumb as the brake (preventing you from bashing the slide into your chops). Add the tongue at the halfway point, keep the air stream even, and you have a beautiful slur!
Holding the slide as described above, you may find the ends of your fingers striking the bell as they go by. Third position happens to be quite close to the bell: should you use your fingers to find it? I think it's OK for student players (who haven't yet developed great intonation) to double check third position with their finger tips. First position is of course all the way in, which puts second midway between first and the bell.
Fourth position is usually fairly easy to find 3 – 3½ inches below the bell. Sixth position, for a person of average size, would be a fully extended arm, with the slide brace held in the normal position as described above. To get to seventh, keep that fully extended arm and straighten out your hand, as if you are about to shake hands with somebody at arms reach. Careful: don't drop the slide! Hold it using finger pressure on the lower slide tube.
Players with shorter arms may find that straightening out the hand isn't enough to get them all the way out to seventh. It's permissible to let the right shoulder move forward to get the extra reach.
Fifth position can be the student trombonist's nemesis. You play in fifth to get notes like F# or Db, which aren't in the typical band keys of Bb and Eb. There's no easy way to find fifth position other than the fact that it is midway between fourth and sixth. It can be helpful to play while looking in a mirror, to see if you are actually putting the slide where you think you are putting the slide.
Whatever you do, don't "fish" for the notes. Go to the position and hold that slide steady until you know for certain whether it is right or not. It doesn't help to make a lot of rapid little adjustments up and down. Far better to let the chips fall where they may, choose a spot and hold it for awhile. If you hear beats in the sound you could be in the wrong place, but it's also possible that somebody else is in the wrong place. Try moving the slide slightly one way or the other to see if the beats slow down. If they do, you did the right thing and can continue in that direction. If they don't, go the other way! This kind of cautious adjustment should be done very consciously and with a minimum of guesswork about slide placement.
Brad Howland is the Principal Trombonist of the Victoria Symphony Orchestra.