Long time since I played, but still... welcome Commander o7
There is no campaign. There is (I assume this is still the case) some background story developing (via in game news articles) There are/were community goals which are a at least couple of days long missions basically that essentially give the community options to branch this background story in a way (depending on if the goal succeeds or fails). Sometimes they used to do two goals at once that may work against each other even. Not quite sure how they manage it now, but I'm pretty sure they made it more dynamic with Thargoids (aliens) fucking shit up. Mostly it's make your own fun kind of game.
There is multiplayer, I assume people still play in Open play. But people also can play Solo. The chatter you see is mostly NPCs, players will have "CMDR" before their names and have hollow (or is it solid? Again, long time :D) icons on radar. It's a big galaxy so might be a a while until you meet other players, unless you go to popular places.
Mouse will give you better control of the ship (or so was the consensus) but I found it infinitely more enjoyable with a joystick or at least a gamepad. HOTAS will give you ultimate immersion - and immersion is I think Elite's strongest suit.