Are modern forks compatible with older style bikes? I have an old 10-speed bike (likely 80's vintage) with a threaded fork. Can I replace it with a modern threadless fork with the style of stem that slides over the top of the fork and clamps on? I know I'd need to replace the stem and handlebars too and am happy to do that.

What do I need to look for? Is the diameter of the head tube different with newer bikes?
The technical term is "threaded fork". That is, the steering tube has threads on top for the headset to screw onto. The newer design is called "threadless" - that's the one where the stem clamps onto the steering tube with pinch bolts.

You can switch from one to the other as long as the diameter of the steering tube is the same. (You'll obviously have to use a headset and stem designed for the type of fork you have.) The problem is, the diameter is often not the same. Around the same time as the threadless forks were introduced, they started making the steering tubes and head tubes thicker. The old ones were typically 1 inch (that's the steering tube outer diameter). The new ones are 1.125", sometimes 1.5".

You should measure just to be sure, but I bet you have a 1". It's possible to get a 1" threadless fork, stem and headset, but the choices will be pretty limited.