I know Elon Musk is shooting for a crewed mission to Mars sometime in the 2020s but what will happen from there?
Elon Musk is doing a Underpants Gnome style mission.
He plans on building the railroad to colonize the West, and others will come along and sell/build other needed infrastructure. Without the Conestoga wagons or railroad, it won't happen.
So he is doing Step 1: Steal Underpants. Ooops, I mean, build a transport system to Mars.
Step 2: ????
Step 3: Profit!
But more seriously, someone has to build the way there; that is his focus. He expects others to step up and create other stuff once they get there.
It does not seem to be seriously more considered than that, at least in the public literature.
If you want to look at the outlined plan I'd recommend watching Musk's presentation at 67th International Astronautical Congress in 2016. You can find the slides presented here.
I'd continue to outline his plan but the video covers all the bases. Some things to note is that SpaceX is currently moving toward the Mars mission by reducing launch costs, improving and leading the way on rocket reusability, developing new crew capsules, improving propulsion systems etc. It is worth noting that SpaceX is currently using its own capital made through its launch services to develop some of this technology. However, the brunt of the Mars colonization expenses will be dependent on the US government whether initially or sustainably throughout the colonization.