I can talk best about what I do.
This is only one method.
There are lots of methods.
To figure out where to begin, I start by looking at the big crisis that shapes the whole story. The Climax. The Big Kahuna.
This comes late in the story, naturally, but a long sequence of protagonist thoughts, actions and decisions lead to that Dramatic Climax.
Now, of course, there's lotsa other stuff that led to that Crisis. Various characters did this and that and weather happened and a declaration of war and Susie came down with the pox.
But in finding my beginning, I'm only looking at what the protagonist did.
That's because the book is about the protagonist and I use her to shape the story.
So ... I go looking at the protagonist's actions.
I sort the actions into two piles:
Those actions that are absolutely necessary for The Crisis to occur just the way it does. This is 'story action'.
Those actions that could be changed or eliminated without affecting The Crisis. Not 'story action'.
Entering the vampire cave is necessary. It is 'story action'.
Buying eggs at the market -- not so much.
I ignore everything that isn't 'story action'.
I go through the story action, decisions and so on of the protagonist, looking for 'tipping points'.
I'm going to use one of these tipping points as the first scene in the story.
"'Tipping points'?" you say.
Ok.
Every story has many places where a different decision would lead to a different Final Crisis. These are 'tipping points'. These are moments where the protagonist has an opportunity to change the story. Her decision could go either way and what she does matters.
Tipping points make the protagonist strong and interesting. Story is not about what happens to the protagonist. Story is what the protagonist decides to do with what happens to her.
And the tipping points are her decisions.
We, as writers, try to show these decisions, because they are so important. Even if we know the protagonist will not run from the upcoming battle, the reader still wants to see that decision to stay and be brave. She wants to see the 'tipping point' where the protagonist has a choice and makes it.
So I look at where we got tipping points.
I want to start the story with a tipping point.
Which one?
-- The tipping point I choose to start the story should hit early in the sequence of story actions.
But not too early.
There's always a long history of protagonist actions, stretching way into the past and extending far into the future. Not all of this important action can happen 'on stage' in the book.
We want to scoop as much important story action into the book as we can. We want all the good stuff to happen 'on stage'. Essential story action that happens before the beginning has to go into flashback or backstory of some kind or prologue. This is a weakness in the story structure.
Note that I'm talking only about 'story action'. We want to scoop in the important stuff that happens, write it out as scenes, and put it on stage. We don't want to put in scenes that have no story action and only convey information. We especially do not want to start with one.
-- Ideally, we want to start with a tipping point that is chronologically tight with the rest of the story. That is, we do not want an outlier in time. This avoids a hiatus in the action. We don't prologue, 'Two years before'.
(Prologues are necessary in some stories, but I feel it may not be the tightest of all structures.)
-- You generally want a tipping point where the protagonist makes an active choice. He has the capacity for some sort of action.
Sometimes, however, there doesn't seem to be much the character can do. The protag is a victim trapped in the dungeon. A Cinderella bound to her stepmother's will.
What the writer does is give the protagonist some choice. That limited choice can be the tipping point.
In the dungeon, she makes a decision not to reveal her secrets, even if it means she has to commit suicide. She gives half her bread to the other inmate -- the man who will later be the key to getting him out of the fortress. She figures out how to blackmail the guard. She palms a spoon to scrape through the rocks.
Or she decides to approach each daily, demeaning, unfair household task with a stalwart heart and a generous spirit.
What's important is the decision.
-- It is especially exciting if the tipping point is a decision that rests on a knife edge. Where it could go either way.
-- All else being equal, I like a tipping point that involves 'big screen action'. That is -- the protagonist makes a decision that results in big bodily motion. Getting-into-a-car-and-racing-down-the-mountain decisions, rather than stern conversation or hiding the jewels in the sofa cushions.
This is just me. Personal preference. Ignore it.
-- You want a tipping point that reveals one of the most basic aspects of your protagonist.
-- A tipping point is always 'story action'. The scene is about something that changes the outcome of the story.
I can't say this too strongly.
The tipping point is not description of an average morning. We don't have two people talking about the lost gold mine.
-- If you can manage it, you want this tipping point to be important. If the protagonist had taken that different path, the crisis of the book would have been entirely different.
-- The tipping point, itself, has to occur somewhere in the first scene, but it need not be laid out right in the opening.
The decision itself may be quite a small part of the scene. What matters is that without this decision, the scene would have gone very differently and the story as a whole would be greatly changed.
-- In genre Romance, one or more tipping points are likely to arise in the hero/heroine meeting.
This is a 'neat' tipping point to start the story with because it ties up several plotting thingums the writer has to do in the first couple chapters.
So, in genre Romance, a good beginning is a tipping point of, for instance, the female protagonist's decision on how to deal with the hero . . .
'Will she rescue the hero or leave him in prison?'
'Will she approach this dangerous man on the street, or find some more cautious way to accomplish her ends?'
'Will she stay to spy on the approaching stranger on the chance he is the messenger she's waiting for, or will she run to safety?'