Mars Horizon

Fairly simple requirements. After you complete the Human in Space Milestone Mission at Earth, in the Requests tab on the Mission Select screen for any planet, look for a Request Mission with the Publicized mission modifier under the description and above the rewards. The icon looks like some microphones. The mission also must be a manned mission, so the description of the mission should indicate the need for an astronaut. Request missions are random so if a manned publicized mission is not available, complete other missions and/or advance the time and one should eventually appear. Once you see a Manned Publicized mission that you want/can complete, accept the mission request.

First you need an Astronaut Training Facility, which is an Era I building that you research on the Research Screen, building tab. You need this building to complete the required Milestone mission so you will already have it. If not continue down the milestones and research tree and eventually you can build it.

Next step is to hire, if not already, an astronaut from the crew screen (left on the D pad down to the crew icon that looks like 3 people.) You are looking for an astronaut with the Celebrity modifier. In my game this was my first astronaut but there is likely some randomness to the astronauts available for hire, if a celebrity isn't available, more will appear for hire over time. Hire said celebrity astronaut.

Then after the capsule and rocket have been built, on the launch preparations tab, select the celebrity astronaut as crew and complete the mission.

Because of the wide variety of possible mission combinations and phases, no guide could cover every possibility so you are on your own to complete the mission, but you have to have completed the Human in Space Milestone mission first so you should be able to complete a similar request mission.