I liked how the DM started things off on one campaign I played in

We agreed that we'd be an already established party so we could pretty much jump right into things and not have to do a whole "I'm putting together a team" thing the first session.

We had our characters mostly ready beforehand and gave the DM a brief overview of them. It was loosely set in the forgotten realms, so we were already pretty familiar with the setting, but if it was a home brewed setting all we probably would have really needed was a brief primer of the major factions and locations. Most of us weren't tied to any major factions, or our ties were pretty loose.

For our session zero, he basically had us figure out how we knew the characters to the left and right of us while he pulled each of us off one by one to run through a quick little backstory roleplay to give us a couple personal plot hooks we could latch onto

For example, I was playing a fighter with a soldier background, I remember the guy next to me was playing a barbarian, so we decided that sometime in the past his tribe had tried to invade my city, hijinks ensued, and we had ended up stranded somewhere together and bonded, missed out on most of that war, and became friends.

And when the DM pulled me back, we ran a quick scenario from another conflict I'd been involved in against cultists of some kind where I'd been captured during the final battle, the enemy had attempted to turn me into some sort of living statue servant, but the spell was interrupted, leaving me with a stone fist. I was rescued, managed to be present when the big bad was defeated, and was counted as sort of a minor hero of that war.

Unfortunately that stone fist kind of ruined my plans of retiring quietly to be a farmer or something (hard to do farm tasks one-handed) so I turned to being an adventurer to make ends meet. Recruited my old barbarian buddy who brought another of his friends on-board etc.

And once we all had our backgrounds and how we know each other sorted out, we hopped into things.

I ended up being kind of a central part of what brought us into town at the start of the campaign- my old commanding officer was getting a promotion and had sent me an invite to the ceremony. Of course it didn't go off without a hitch and the ceremony was interrupted by an assassination attempt, and evidence pointed to it being tied to the cult we thought was defeated in that previous conflict, but there were some other things dropped that tied into some of the other players' backstories as well, and off we went.

I think that did a very good job of making us feel like we were part of the world and not just some murder hobos dropped into it to run rampant. We all had history with each other and in the setting, and it wasn't just words on paper that no one else was really going to read, it gave us a chance to actually act out a bit of that backstory and sort of collaboratively figured out our place in the world before the campaign started.