Will it be Starship or SLS? China? NASA and Space X?
Mars will be an International effort possibly led by NASA. SpaceX will be involved as a contractor on the NASA/International mission.
I expect this to happen in the 2040s and by then SpaceX will have a reliable "Starship" system that NASA can hire to move bulk cargo. It Starship ever gets "human-rated" it might even carry people. But after the space shuttle, I doubt NASA will allow astronauts on a launch vehicle with no abort system that works from pad to orbit.
SpaceX has no Mars program. At best they will have a launch vehicle and can offer transportation. But remember, the launch cost is only 10% of a typical space mission budget. So if SpaceX can reduce the cost to zero, that saves only 10% of the total. When you see a Falcon 9 on a launchpad you have to figure the vast majority of the cost is inside the nose cone fairing. The rocket costs less than $100M and the spacecraft being launched might cost $1B.
Musk is working in reducing the cost of the cheapest part of going to Mars.