Select the "Power of Three" mystery when setting up a new game.
This mystery plays out mostly by itself. Cubes will start to appear around your colony. You may wish to build special statues with them (no apparent effect) or just destroy them. A new storage unit for Black Cubes will also appear in your Build menu. You can use these storage areas to have your drones automatically clear the cubes from your build area.
Eventually, you will come across an update about researching the S.H.E.P.I.D. technology. The mystery log will read, "We should research the S.H.E.P.I.D. technology to protect the colony structures from the Black Cubes." At this point, be sure to build a Concrete Extractor if you do not have one. I went heavy with Extractors in the early days of my colony so I had decommissioned all of mine due to stockpiling ridiculous amounts of concrete. However, you need at least one Extrator in order for the story to advance. Once an Extractor is destroyed by a Cube, the technology to search (and therefore advance the mystery) will appear in the Breakthrough queue.
There doesn't seem to be any way of failing this mystery, so enjoy the story as it unfolds!