The core of my story is the relationship between two childhood friends, with magic as a connecting element. In order to set that up, I need at least some background on magic and mages in this setting, as well as a bit of setup for one character's archaeological work (she decodes and determines the uses of magical scrolls found in ancient ruins.) However, I don't intend this to be a very long story, and I don't want to spend too much time worldbuilding when I could be showing character interactions. How do I get the world out of the way quickly, and how much can I get away with not specifying at all?
In many short-stories I have read with similar fantasy/sci-fi settings, the actual world is left intentionally vague. Instead of trying to explain everything quickly, I would advise only saying what is necessary to be said for the sake of the narrative. If the narrative requires something to be explained in order for it to make sense to the reader, do so; otherwise, I wouldn't bother with it for a short story. Write your narrative out in a rough draft, then have a friend read it fresh (or post it here in the forum). Have them mention or mark with a sharpie every time they don't understand what you are saying or what the characters are doing. Once you go back and fill in an appropriate amount of information/world-building, have them read it again (and someone else). Continue this until your reader is satisfied. You would be surprised how little detail a reader actually needs when it comes to this issue. The vast majority will simply be filled in by their imagination.
Most of it. The vast, vast, VAST majority of it can go unsaid. Magic exists? Check. Archaeology exists? Check. Ancient scrolls? Check. Done!
How much does the reader need to know in order to understand your plot and characterization? That's what you need to present. And don't overestimate how much they really need to know!
Magic is one of the hardest things to explain, at least that is how I see it. That said, I think you need to leave lot of it open. Create sentences that give a small assumptions to the reader about the logic in your magic. Character 1: "Isn't that the stone with burning glyph in it?!" Character 2: "I think it is!" One sentence and you can already wonder your thoughts about the relationship between the magic and these stones. Hopefully that example gives you tips what I mean. I wouldn't go much deeper than that.
Leave it vague. Characters can 'show' their world without it having to be fully explained and fleshed out. That magic exists can be shown with a simple spell. That someone is a mage from being called: Mage. The archaeological work could be explained away and shown as something tedious, a life's passion, or just part of what goes on with a bit of dialogue.
I recently completed my second draft of the story. It came down to around 3,800 words, of which about 350 explained the setting in some way. For instance, this was my whole explanation of Josie's job and current motivations: -- -- -- -- Josie plopped back down in her seat with an audible thud and gestured for Nikki to join her. "We've been digging up so much stuff in the Jadehill ruins. Tools, sculptures, even weapons. I've been focused on the spellbooks, of course, but a lot of that's similar to what we already knew. Magic's come a long way in two thousand years." "You found something new, didn't you?" Nikki guessed. "Something that might go wrong if you tried it on someone with magic resistance." "We still don't know exactly what it does," Josie explained. "It has something to do with sharing the caster's physical sensations, but there's a lot more to it than that. All we know is that it says never to use it on an animal, and never, ever to use it on something that isn't alive. Synchronizing two people's magic is hard enough when you know what you're doing, and with a spell like this, almost anything could happen. The first test has to be someone who won't resist it and leave it half-finished."
Yeah, that works. Josie is explaining something to Nikki, but because Nikki doesn't already know it (nor does the reader), this device works. Josie wouldn't stand explaining something to a person whom she knows already understands the information, would she? You also kept her explanation brief and clear. It works fine. Keep on the way you're going, and you'll be fine.