Compared to what you've gone through to reach this point, this final story mission is a cakewalk. It might be of little surprise that your visit was not unexpected. The Institute's leader, simply known as Father, will guide you to Shaun and will answer many of your questions in person. The conversation will conclude with Father asking you to join The Institute, which is needed to complete the quest. You can rescind your allegiance later.
The final steps to this quest is to introduce yourself to key staff of the Institution. You'll find the entrances to many of the labs along the walls of the main concourse. You can find Doctor Clayton Holdren in the Bioscience lab.
To find Allie Filmore, explore the main concourse.
Enter the Synth Retention Burean to find Justin Ayo.
You can find Doctor Madison Li in the Advanced Systems lab. One bonus of meeting her is that she'll modify your Pip-Boy with the ability to fast travel in and out of the Institute.