How do you start your new campaigns?

Typically, something like Fate's phase trio: https://fate-srd.com/fate-core/phase-trio . The tldr is you establish history between the characters before you start.

I don't really like the "you are working together because you're both players at the table, not for any in-character reason" trope at all.

The example you give in your OP is not a good idea for an interactive storytelling game with friends. It has too many spots where the players might want to do something, and you're going to want to reject it because you have a set idea of what needs to happen. Either take this idea and move it to your "I should write a book" folder, or start the game where it has already happened.

You can totally start a first session with "You were hired by Dmitry Peacile to do a job. The whole time he was aloof, arrogant, and insisted things be done his way while you stay out of the way. It became apparent you were only there for a technical reason in the contract, even though he kept the paper on his person and never let you read it.

Well, just a few minutes ago he rode his horse over a rickety rope bridge while insisting you all wait here. The bridge gave way, Dmitry, his horse, and the contract that entitles you all to a big payout plummeted into the depths of Goblin Song Gorge below.

What do you do?"

Now you're starting where the players have some agency instead of possibly wasting hours doing stuff where you don't want the characters to change things.

I may be old fashioned, but I love to start in a tavern. It's a place that can have a lot of npcs hanging around that can be introduced and then reappear later in the adventure.

Usually I prefer to start with the party already formed, or have the characters have a connection between each other from before the start of the adventure. Imo it speeds up the initial stages of the game and gives everyone a preexisting reason to be in the party.

I had some pain in the past with players that didn't want to find a reason for their character to join the party, and asking them to have one as a prerequisite can help to filter too mich edginess from the scene.

I also like to start with combat or some other dangerous situation. I start with some talking and a breef introduction to the aim of the adventure, then have something unexpected interrupt the talking, a fight, then back to the talking.

Totally depending on the game/setting/campaign genre. I see the classic you're in a tavern as a bad practice and try to avoid-it (OSR may be an exception). In general, I would address the group consistency/relationship/and why they want to work together as part of character creation, but sometimes I'll start in media res by breaking the PC relax life with a dramatic event

I like to start my party off immediately after they failed their first quest. Session 1 is explaining the consequences of that failure, what that quest was, and how it went wrong.

For example: "Two towns hold us responsible for the bridge that collapsed in our fight with the troll, who escaped, because we didn't burn the body. We're at worst fugitives, and at best in extreme debt. The adventurer's guild is pissed that we failed, too, and is demanding their cash advance back."

I usually have the party band together as part of their backstory, so that we don't have to roleplay for the upteenth time the characters getting to know each other.

But if we start as strangers, I don't see anything wrong with the usual tavern or festival. The first session is a bit critical from a roleplay perspective because everyone is playing new characters and is focused on getting the "feel" right, so I don't mind easing the job for them by placing them in a well known and established setting such as a tavern: in this way, they can relax and think less about their surroundings, and more about their characters.

I like to start with some kind of action that also gives the group a reason to work together. Eg the inquisition drags you all out of your beds in the night and chains you together, or (in the game where everyone was a werewolf) you're out in the woods hunting and this deer can run faster than you, how do you work together to take it down, or you've all been pressganged to work on some evil bastard's ship, what do you do about it. That kind of thing.

I could see a party abandoning the quest to try and go after the DMPC's stuff, if it's cooler/more valuable than the quest reward.

Maybe have it be an old soldier that lives in the town, and the townspeople are trying to get him to stop trying to handle all the problems nearby alone because he's getting old. So his gear would just be his old military stuff. Good, but not worth diving into a river for. He's grumbly about having to sheperd some runt adventurers so everyone will stop pestering him, and insists on scouting ahead so the youngins don't just stumble into an ambush and get their heads cleaved off.

Cue rope bridge.

That also gets you potential hooks with the town not trusting the party since the guy didn't make it back, or being understanding and throwing a wake/funeral.

Or hell, if the party succeeds some checks and keeps him off the rickety bridge, you don't need to worry about him hanging around past the first "quest" messing with balance. Him missing the signs of the rickety bridge gets through to him and he decides to retire.

I once started a Star Wars campaign with the players meeting in prison. Most arrested by the Empire for things they didn’t do. A couple for things they did.

First episode was breaking out. I planned out a few options and dropped hints. They of course chose brute force. At the end was a choice between a few different ships to steal and get off the planet and use for the rest of the campaign.

If I am planning on doing a sandbox in a modern setting, I like:
Everyone was invited to NPC's 40th birthday. This NPC is a bit of a social butterfly, and has invited friends from all of their social groups. I let the players know that they can either choose to already know each other or not, and they are likely to have at least heard a positive story or two about each other. Then inciting indecent happens at the party, or the police come because they are investigating the mysterious death of the birthday NPC.

If I am planning on doing a sandbox in sci-fi or cyberpunk:
Players respond to a Craiglist-esque easy-money job listing; they can choose to have known each other beforehand or not. Things go wrong, and they are left in possession of a small spaceship and a grudge.

Not a fan of You Meet In A Tavern but I do love You Meet At A Festival instead where you all explore what your characters do at a party before, y'know, zombies attack or whatever

My setting has a town, kind of like the one at the entrance to the Grand Line in One Piece, where most people going on an adventure have to stop. The Townspeople there know adventurers do better in parties, so they guide any visitors to the local tavern to help them form groups, and so that they can get gossip about them, as nothing much else happens in this small mining town. The tavern is run by a former adventurer that tells them about the first choice they're going to have to make in their journey, which path to take through the mountains.

So it's not me that was pushing them together to make a party out of nowhere, it was nosy locals in setting. This also let me introduce a recurring rival character who refused to join their party because he's a rich asshole.