I'm going to use the openings supplied by commenters as examples of how different openings lead the reader into anticipating a different type of story... even if the basic elements are the same.
Here's my scenario:
Let's start with an easily adaptable plot. Sarah, a young woman, goes to the funeral home to deal with the necessities of burying her Uncle Wally. She realizes somehow he was actually murdered, though everyone has assumed the cause of death was natural or an accident.
So here are some commenters' takes on that. I'm just going to use the first line or two of each to show how even from Line 1 you can get the reader's anticipation up.
This is Keri's:
Sarah Smith gasped and turned around after walking in Uncle Wally’s house and finding him face-down on the floor with a knife in his back.
I "diagnosed" this as romantic suspense, but it could just be suspense (no romance) or it could be a fairly standard mystery. Why suspense? Because Sarah is the one who discovers the body, and it happens very early. A mystery might have the "sleuth" discover the body, but that tends to happens -after- a set up of the setting and situation-- several pages, maybe even several chapters (the English mystery often puts the murder in Chapter 3 or so) in.
Also notice that Sarah is closely related (niece) to the victim, which again is more of a feature of the suspense novel. Suspense, as its title implies, is meant to be a more emotional experience, while mysteries are more intellectual... so the more emotional experience of the close relative being killed is "suspense-y".
I would also point out that naming a character early conveys personal importance on her. That is, naming Sarah Smith in the first line suggests that she's going to be the sleuth.
NOT naming her until later would indicate to the reader than she is not going to be the main character-- maybe she'll be the victim, or maybe the murderer, or maybe a secondary character... but if she's not named on first reference, that's usually a sign of secondariness.
If the character is named by her -role- (The waitress fiddled with the spoons), that sends a signal that the role is the only important thing about her.
Now obviously, you can play with expectation here... but you should probably know what the expectation is. When you are a reader, for example, you might find a bunch of names in the first couple paragraphs confusing because you're subconsciously searching for the protagonist-- the one you're supposed to identify with. So when you re-don your writing hat, don't forget the experience of the reader. Don't feel like you have to name the doctor and the cop and the neighbor, especially if they're never going to appear again.
So Keri using Sarah's name and Uncle Wally's (AS Uncle Wally, not as Wally Magnuson-- that is, identifying him from her perspective) tells us that these two will be important... but that Sarah (both names, plus of course we're in her point of view) is the main character.
One little editing note-- if I were editing this, I'd point out that you have reaction (her gasp) before the cause (seeing Wally dead) in the sentence. How can you rewrite that so that you get a more logical sequence but still have the "surprise" of the murder at the end of the sentence?
We think chronologically, or at least, when we assemble our thoughts in a logical way, the order we use for time sequence is usually chronological-- first this happened, and then that. So a reader is going to be just a bit stopped there because the sequence in the sentence defies the chronology. It's not a big problem, but you want the first sentence to be perfect-- because readers can so easily put a book down at that point, as they've got no investment at this point. Don't give them a reason.
Also notice that the focus on action rather than perception (her gasping then telling us what she's seeing) is presenting this from the outside, from the camera eye. The great advantage books have over movies is that books can be INSIDE the character, while even the most deft camera operator has to remain OUTSIDE. Telling the action rather than letting the reader FEEL the experience of being Sarah is a point of view choice-- you're choosing "objective POV" rather than the more personal, interior "deep-third-person". That is not unusual in the first paragraph of a book, but it tends to put it more in the "thriller" genre than either suspense (which is all about the reader sharing the character's experience) or mystery (which is about the reader sharing the sleuth's thoughts). So if you want this to be about Sarah as a potential victim (suspense) or the sleuth (mystery), I'd suggest a closer point of view from the very beginning.
So you could fix both of these problems by going inside her and narrating her experience rather than just her action. Start with some emotion or thought word that places us squarely in her POV, then narrate from the inside, like--
Sarah Smith wanted to turn around and walk away without knocking. She didn't want to meet her Uncle Wally. She had enough troublesome relatives already. But she reminded herself that family was family, and knocked.
The door wasn't locked. It wasn't even latched. Just her knock was enough to make it swing open. Sarah was confronted with Uncle Wally's front hallway-- and there, framed in the sunlight from the open door, was Uncle Wally, face-down on the floor with a knife in his back.
Much longer, notice-- deep POV is always (almost) longer than objective. But notice that it's her experience from the inside, so from the beginning, the reader experiences HER in real time.
Must run-- more later.