How can you put a lot of time-skips as soon as possible in a story? Because of certain restrictions, I am forced to put a lot of time-skips at the beginning of a story, but how do you do that effectively while keeping your readers engaged, not confused, and causing other negative impact? I tried to spread them throughout the story, but the best parts are at the end of the story, and I need to get people hooked to the story, so because of that I need to put a lot of time-skip scenes at the beginning of the story. Rewriting the whole story is not an option due to deadlines and budget.
Now, I don't know if you would consider my suggestion within the 'rewriting' domain, but if "the good stuff" begins later in the story, perhaps the story itself should start closer to those parts.
One way to do this, if you still want (or need) the 'early parts' included in your story somehow, is to present them through flashbacks, or, as I personally prefer, through dialogue or 'showing' when following your characters - or through material otherwise being presented in the story.
I haven't worked with flashbacks myself, so I will focus on the other options.
Through dialogue (and behavior), we learn a lot about your characters; what they're like and why (what they've been through and how their personality and abilities played into this).
This could either be through dialogue with other people who were present at "the important earlier parts"; People have a tendency to visit past shared experience as a way to maintain their identities.
It could also be with people who weren't there; When we get to know people, what they mention about themselves, and how they present this, how they react, what they choose to tell and not, shapes our impression of them, becomes their identity to us. Your characters doesn't even have to explain these things themselves - We could 'overhear' a fellow character explain "why X character is like this..." e.g.:
"Oh, and btw., don't ask about the scar.. long and sad story, not enjoyable for anyone here."
'Really? What happened?'
"Some horrible war stuff, lost a lot of family.. Gruesome shit."
Perhaps your characters interact with enough known/foreign characters that the "important earlier parts" can be presented in a way that doesn't force us to experience them 'live' if it isn't actually that interesting.
Alternatively, we may be presented with "footage" (e.g. from cameras) actually 'showing' parts of the past. Here we may also have the opportunity to experience other characters reacting to or discussion what is being shown or presented.
Time skips exist to cut out any information that either does not benefit the plot or that the audience does not need to know at this point.
Let's say this was a superhero story, for example. The hero gave up their crime-fighting and went to work at an office job for three years.
Unless there's something important happening in those three years, no one wants to read about the boring office job. It makes perfect sense to cut it out.
Alternatively, the skip is intentional to keep the audience in the dark. Perhaps the character was knocked unconscious. The world was fine when they were awake, but it went into absolute chaos and now they have to figure out why. That adds some intrigue. The audience will want to know what happened in the interim.
Time Skips need to take into account three things: Quantity of skips, Pacing of the skips, and Length of time being skipped.
Quantity-The more time skips you have, the more confused your audience will be. If you are constantly jumping through time things get complicated and the audience loses a sense of grounding.
Tommy starts as a smiling, goofy eight-year-old but...snap He is now a moody teen. snap He is now a depressed adult. snap He is a bitter old man.
Do you see how that would give an audience whiplash?
Pacing-The faster these skips are the more confusing they are to the audience.
Imagine eight-year-old Tommy riding his bike with his mom, having a good time, and then...Welp, now he's eighty.
No preamble. No preparation. He's eighty now. Deal with it.
Abrupt and jarring changes are fantastic (she said with sarcasm).
Length-An audience is generally more willing to buy smaller time skip. If it's a simple jump of a day or maybe a week and there was plenty done in that time, it's perfectly understandable.
But skipping a whole month? A year? A decade? Then things start to get crazy.
Massive time jumps also leave lots of room for author slip-ups, plot holes, and excuses.
Authors can and will use time jumps to excuse almost anything, and it's usually not a good sign.
Why is the character acting wildly different? Time jump. Why are there new characters we never got introduced to before? Time jump. Why is the plot completely derailed? Time jump.
Always blame the time jump for all your problems...If you want the audience to dislike you, that is.
If they really are at the start of your story, put the earlier material into a Prologue which can be a series of earlier short scenes (one per main character maybe, to show what each was doing when they first heard about the magic McGuffin). By definition a Prologue is an introduction so covers something the Reader needs to know, but that may not really be part of the current action (or the "best parts" OP refers to).
As a reader, I know a Prologue is often followed by a giant time-skip so it doesn't disrupt my reading at all.