Starting Session, skip the "getting to know you" part, everyone knows one another.
So you're asking, okay, well then how would I get the characters to know one another right away?
Its simple, when they are rolling up the characters, ask them to provide a reason they would associate or know one another.
It could be as simple as "my character's father was a baker and I delivered bread to blah blah's character's family, I knew of his character that way."
It could be as complex as they lived next door, they played together as children, then as time passed one family moved away and they had not seen one another except for a few times at the local tavern where they had a drink or two, but never had the same relationship they had as young children playing in the meadow.
The other reason I start things off this way, because it already sets the tone of things that I as a the DM mean business and want to get to the adventure and the fun they are looking for in a game.
Heck I've seen a getting the party together session take 3 hours and its really trying on a DM, especially if you have players who like to "be a dick".
The only person at the table who should be the dick, is the DM, its our job. Its the DM vs the Characters.
So next time you think of starting up an adventure, remember all this and maybe you too will skip the "Getting to know one another" part. If you have a good group of players, they will not even care and roleplay their views and thoughts into the story as you form it.