The most important part of a story is its first paragraph. If you don't grab the reader's attention right away, chances are they aren't going to stick around, and writing an evocative and intriguing hook is a sure way to make them turn the page and keep reading.
There are lots of ways to write a compelling first paragraph; some stories begin in media res, others rely upon evocative description to set the tone, or to introduce the protagonist and their dilemma right from the outset, but however it is done, the goal is to immerse the reader in the story immediately. The wrong way to do this is to lead off with a lot of exposition.
This is probably one of the most common mistakes made by inexperienced writers (and sometimes, unfortunately, by experienced ones, too). I would say that at least 80% of the stories submitted to the Libram Mysterium anthology series begin with a history lesson. I think that there is a misperception, especially among writers of speculative fiction, that you need to thoroughly explain your setting and its history before the reader can make sense of it. This is wrong: the reader doesn't need to know everything right at the start, doesn't want to know everything right at the start, and shouldn't be bombarded with an information dump when they are still in the 'getting to know you' stage of the relationship.
Imagine you're at a party and you've just met someone that you might like to get to know better. If they immediately launch into a lengthy and detailed narrative of their life history with all sorts of intimate and personal details are you going to want to stick around and continue the conversation, or are the chips and dip starting to look really attractive? It's too much information too early in the relationship. This is all stuff that you'll probably want to learn gradually as your friendship deepens, but learning all in the first five minutes is weird and awkward and it will bore the crap out of you. Instead, when trying to impress a new acquaintance you should try saying something witty and interesting, maybe even something controversial so that, instead of tuning out, they respond by saying 'what do you mean by that?'
Being a writer is a bit like being a stripper. Your job is to entertain by teasing and titillating your audience; you show them a brief glimpse of what lies beneath with a promise of revealing more as show progresses, and it is that promise that keeps them in their seats. If you walk out onto the stage stark naked and shake your naughty bits in their faces they have very little reason to stick around for the rest of show and they might even be turned off by the crassness of it. There is a reason that level of exhibition is left for the finale when the audience is primed for it.
Here is the opening paragraph from a book called Dragonfly by Frederic S. Durbin. It is probably the best opening paragraph I have ever read:
Bad things were starting to happen again in Uncle Henry's basement. These were things that had happened before, when the wind swung round, when the trees all felt the blood rush to their leaves after the exertion of August and the idling of September; when the chuckle-dark harvest moon shaped pumpkins in its own image, brought its secret wine flush to the scarecrow's cheeks; when the rich bounties of the land lay plump for the taking and the light left them alone for longer and longer at a time. But when the trouble started before, I was too young to remember.
This paragraph is pregnant with promise. The very first sentence, 'Bad things were starting to happen again in Uncle Henry's basement,' is a powerful hook. We know that bad things are happening. They are are happening in Uncle Henry's basement, and they have happened before. Those first eleven words are loaded with implication; immediately you want to know what the bad things are. You want to ask Durbin what he means by that. You want to learn more. Durbin has hooked you. Then, while you're still wondering about that he hits you with some evocative imagery that not only sets the scene but also hints at the nature of the bad things, which we can guess take place around Halloween. We do find out in the subsequent pages what happened in Uncle Henry's basement years before, which is great because by the time Durbin tells us we are already invested in the story and are dying to find out. If he'd begun the story by telling us what happened in Uncle Henry's basement we wouldn't have cared and probably wouldn't keep reading.
Handling exposition in a story can be tricky; the information is important, but you need to exercise restraint when doling it out, just like the person at the party. It is better to reveal information little bits at a time instead of dumping it all at once, which can overwhelm the reader, and as a general rule it is better to show the reader the information than tell them. This can often be achieved through dialogue or activity, but be subtle. Don't resort to the heavy-handed 'as you know, Bosworth' type of forced expository dialogue that lazy television and movie writers inflict on us. Reveal the information slowly and naturally and don't be afraid to let the readers fill in some blanks for themselves. Part of the fun of reading a story is figuring out what is going on from the hints rather than being told.
But however you decide to write exposition remember these guidelines: 1) not too much, 2) not too soon. And whatever you do, use your opening paragraph to entice your readers, not to bore them. Be the stripper; get us worked up and aroused by taking your clothes off a little at a time instead ripping them all off at once.