I am going to switch out my quill stem for an adjustable quill stem. It looks like I have to make sure of the type of stem, the stem diameter, and the bar clamp for the handle bars.

I think I have a threaded (no spacers, cap over a lock nut and the quill with wedge in the tube)
The diameter of the current stem is 1"
The handlebar diameter is 1 inch

So I should get a 1" (25.4mm) adjustable quill with a 1" (25.4mm) bar clamp?

Are you absolutely sure?
a 1" quill means it is a 1.125" fork steerer. 28.6mm
the headset will use a 36mm wrench,,
If it is a 1" headset , that is the steerer tube OD.
the Quill is fitting the ID, and that is 7/8" aka 22.2, metric equivalent.
the wrench usually used is then a 32mm

Yea I'm not sure that's why I am asking. It would be easier if you could just measure your current part and that are the measurements to look for, sigh. all I know for sure is the current quill stem is 1" in diameter, when I remove the quill stem the receiving tube hole is 1" and the handlebars are 1" in diameter. I'm not sure how the listings of stem sizes relates to those measurements.

If it is in fact a 1 1/8" threaded fork , think about a quill to threadless adapter,
then you will be able to buy a stem to go on the adapter,
from the more abundant threadless stem sector.

So it looks like the "size" of a stem is actually 1/8" bigger than the actual diameter of the stem. Maybe the tube is 1/16" thick? So if your stem is 1" you need a 1 1/8" adjustable quill stem with bar clamp to fit your handle bar size.

headset is dimensioned for the outside, quill stem slips into the inside

tube wall thickness, is 1.6mm , in both cases.. half of an 8th is a 16th.

pick up a decent caliper, PRC makes decent measuring machinist tools too.
digital LCD ones have a button .. Inch/MM..