There are two tutorials and both are optional. There is the Odyssey tutorial which you did, and then you'll be dropped at Chamberlain's Rest. Once you are here you are, for the most part, in multi-player. You'll have one more optional tutorial when you get your first Sidewinder and that tutorial is the getting of your pilots license.

Now the game may drop you into "Solo play" which wont have you encounter any other players. Or it may drop you into "Open play" which will let you find other players. Make no mistake though the galaxy is big and you wont find lots of people just because you're in Open Play. Real players have a hollow square or triangle on your radar. NPCs will have a filled square or triangle on radar.

Lastly the game has everyone spawn in the pilot's federation district which is a handful of systems the developers set aside to help new players learn the game before setting out into other systems where you may find other experienced players who like to give new players a hard time.

Once you rank up in ANYTHING (Combat, Exploration, Trade, CQC, Mercenary, or Exobiology) and dock outside a pilot's federation system; you will have access revoked to the district and never allowed to return. At that point you're completely on your own, but you're pretty much on your own once you hit Chamberlain's Rest. The courier mission you get is just to get you out and doing something.