What is the proper etiquette for requesting a meeting with a professor who you don't do research with or take a course from?
Say, the professor is doing something you're interested in, and you just want to talk to them just to learn more about their research (you might be interested in their research out of pure curiosity, or might be wanting to find a possible connection between their research and your potential research). How should one phrase the email?
In particular, I'm interested in trying to find a way to set a particular time for a meeting. Sometimes I ask "are you available next week?" and they say they are, but it's possible that they could be away for most of the week (for unexpected meetings and the like). Should I request a time of my own in a reply? (especially if they don't give a time).
I'm assuming that you are a graduate student.
Simply send a short, polite email expressing that you are interested in his/her research and that you would like to talk about it. If you share a coffee machine, lounge, canteen, etc, you could suggest to talk over coffee. Certainly, say who you are and what you do in the email, and indicate why you are interested.
Don't ramble on in the email. No one has time to read long emails.
Showing interest in research and showing that you won't be wasting his/her time are main points to observe. Most researchers (unless they are ridiculously famous and super busy) would be happy to explain their work and create new avenues for future exploration.