When I was an alto sax player I’d always start by playing a G and holding that a bit/adding vibrato to check the quality of my reed, then go into a G major scale (concert C). Once I learned the (major) circle of fifths I used that as a warmup every day. Unless it was jazz band practice, in which case I’d play a few harmonic minor scales (usually starting with G).
I haven’t touched a piano in ages but the first thing I invariably did was play Heart and Soul (by myself, muahaha). Then a jazzy version of Heart and Soul.
I don’t do anything specific anymore other than start with with a second inversion chord in the right hand and 1-5-1 in the left. I used to would do a massive rolled chord of whatever key I was starting in. I usually save that for later now that I’m out of practice and need a warmup.
Keyboards: it depends on the board and if anyone’s listening.
I learned from a Richard Tee video the “trick” of playing some really impressive-sounding little two-handed flourish when sitting down before playing in front of others – very short, but since my MO when playing is to start simple and build to something organically, this sets people (especially other musicians who might be joining in) at ease that you’re not just going to be trinkle-tinkling the ivories, but radiating the 88s if need be.
By myself, I just play a few chords, on piano, usually a gospel Ray Charles or Billy Preston or James Booker type thing, or maybe some standard like “Blame It On My Youth” nice and easy; on Rhodes, usually some tune from a Herbie Hancock or Patrice Rushen or Chick Corea thing; on Hammond, usually some fast blues runs over a medium bassline or a 6/8 thing. It really just depends what I feel like at the moment.
At the piano, I usually begin with “Lean on Me.” I have no idea why.
On my C flute, I’ll begin with a G, and hold it for a while. Then the same for some other notes, randomly chosen. Then some scales and the respective arpeggios.
The gear part is less interesting than you’d expect, WM. And most of the tunings are “normal”; open D, double-dropped-D, open G, etc. But yeah, I’ll get around to it.
On guitar I strum the open strings to get an idea if it’s in tune. Then a few first position chords to get a feel for the string tension.
On piano (which I don’t play but can usually recognize by sight) I play a C chord, Cm, pause, Cm, C which are the opening chords to (among other things) “Thus Spake Zarathustra.” Since that’s almost all I know, I nod sagely and go off to do something else.
My thing used to be after tuning was to run through the opening of John Lee Hooker’s “Boom, Boom” which combines a simple warm up riff along with some big, ringing open chords to make sure I really was in tune.