I learned this exceptionally useful skill early on, and sometimes I take it for granted. From ages 13-18, I’d roll out of bed every Sunday morning and play the church service cold, not even looking to see what the hymns were until the prelude was underway. Practice makes second nature, and that’s served me well at many a gig.
Several different people tried to teach me transposition by several different methods. For those not in the know, a standard Bb trumpet (or clarinet, tenor/soprano sax, treble clef baritone, etc) has to play a note one whole step above the note you want to hear; it’s just the way music is written (a topic for another day). Other instruments transpose different amounts or not at all.
One suggestion: read the treble clef parts as though they were tenor clef and adjust the key signature. That’s clever, except that I learned to read tenor clef by pretending it was treble and reverse-transposing. It might work for someone else who learned tenor clef first, but I can go years without seeing any kind of K clef so it really doesn’t help me. (Eb saxophone/bari sax players can pretend they’re reading bass clef, with the same assumption.)
Another suggestion: count two half steps before playing each note. Variation 1: count a major second before playing each note, which means constantly shifting your key center. No thanks.
Third suggestion, which is how I started: adjust the key signature, then play the note right above the written note (space above the line, line above the space). It takes surprisingly little practice before it just clicks, and sometimes I habitually do this when the part is already transposed for me. How embarrassing.
Nobody suggested this, but it’s what I eventually started doing: Translate everything into scale degrees (or solfeggio), then translate that into whatever key you need. This sounds like doing it the hard way because it’s a two-step process, but again, after a while it clicks. (Of course, you have to know your scales and keys very well.) It doesn’t work for everything; anything in an unusual scale (basically not major, modal, or harmonic minor), anything with a lot of accidentals, or anything atonal pushes me back to reading the note above the written, at least for a measure or two. However, this approach lets me transpose to any key, which means I can read off a sax part or allow the accompanist to choose a key (after negotiation, because let’s face it, some keys are better for the instrument). It also means I can adjust if someone tells me the wrong key, an accompanist forgets to remove a guitar capo or MIDI transpose, or a singer/choir drifts out of pitch. Perhaps best of all, it means I can use the extension on the piccolo to put me in a key that’s friendlier to the fingers and has better intonation.
What I didn’t mention is that this technique also requires you to be able to put music into and out of your memory on the fly. Remind me to tell you some time about sightreading from memory.