It doesn't stop at the fork.

1" threaded:
Generally uses a more traditional, but also available in modern, 1" threaded headsets.
The steerer tube is generally cut off to work with the stack height of the headset.
The more traditional quill stem is used, and height adjustments are made by inserting the stem farther in/out of the headset.
Bars can be 25.4 to 26.4 in general. 25.4 are most common, 26.0 in a few, and Cinelli bars are 26.4 to match most Cinelli stems.
You have to match clamp size, in general to the bar, or vice versa.

1" threadless:
Generally uses a modern 1" threadless stem, which clamps onto the steerer tube.
You use spacers to set the height of your stem, and often cut it off near your optimum height.
The stem can be flipped, and the stems come in a wide variety of lengths and angles from the steerer tube.
They are now most often sold in black, but some may still come polished.
The stems now are most often 1 1/8" in inside diameter at the steerer tube, so many require a 1" to 1 1/8" shim.
Stems also have a clamp size, generally small (25.4 and 26.0) and OS (oversize), meaning 31.8, to match the bar size.

Traditional quill stems tend to look more at home on traditional steel lugged frames, or those with smaller diameter tubing.
Modern stems tend to look more at home on more modern frames, or t hose with larger diameter tubing, like aluminum.

It's all a matter of opinion. It's a lot easier to change stem sizes on threadless steerers. Changing quill stem sizes requires disassembling half the bar, and often loosening cables, etc.

I have no idea what the bike frame looks like, so I won't offer a subjective opinion on that.