At the risk of stepping into another religious discussion, I'd like to describe how I tune a new horn, one whose tendencies are not yet known to me.
I first warm the horn and my chops up by playing it for a few minutes. Long tones are good. With a good electronic tuner, running, I play an open, in-the-staff C WITHOUT LOOKING AT THE TUNER YET! I play the note until I feel that I'm right where the horn wants to be. I then (while still playing) look at the tuner and see where I am. I then correct for the error with the tuning slide. I repeat the C without looking at the tuner and then when the horn feels good, I look again. I repeat these steps until the C keeps being in tune without my looking.
If you look at the tuner WHILE tuning, you will correct for the errors by lipping them up or down, and negate the whole point of using the tuner in the first place!
I then repeat the above steps on the other open notes, then on the second valve notes, the first valve notes, the 1 and 2 notes, etc.
The key here (no pun intended) is to allow the horn to play in the middle of it's slot before trying to correct it with the tuning slide. I may find that keeping a pencil and paper record of the tendencies of the horn is a good thing to have. In general, however, once I'm familliar with a horn, I rely on my ear to play in tune with those around me, piano, orchestra, band, whatever.
DON'T be one of those people who I see so often with a tuner on their stand all through rehearsal or concert, relying on it to tell them if they are in tune, because if they follow the tuner, they won't be! Some notes have to be flattened, some sharpened, in order to be in tune for the position in whatever chord is being played at that time. My wife knows all of these and can rattle them off if asked, I only know that it sounds right when I'm doing the right thing, and the correction is automatic and nearly instantaneous for me.
Be aware, also, that using a mute will also likely alter your intonation. You need to know what direction the mute(s) affect your sound!
I hope this helps!
South Bay Brass
Silicon Valley Brass Band - A Traditional British Brass Band