This location can only be accessed during the main quest "The Molecular Level". Similar to the Railroad HQ, you can leave the institute by fast-traveling directly. The map marker will become available in your Pip-boy after unlocking this location.
Robotics
Located to the southeast. This is where you'll witness the actual synth production.
Synth Retention
The SRB storage area has fusion cells, institute rifles, and other items to loot.
This division is handled by Justin Ayo
Advanced Systems (northwest)
Bioscience (northeast)
Headed by Clayton. There are two entrances here leading to the FEV Lab.
In the bottom floor to the SE near the central plaza, you'll find the synth requisitions vendor. You can buy the legendary plasma rifle called Experiment 18-A there as well as some unique synth armor pieces.
FEV Lab
In the back passages of Bioscience, you'll eventually reach a Novice door leading to the FEV Lab. Going through this method will put your through three laser turrets and an assaultron. The warehouse has several containers with components and pipe upgrades.
Continue through the hallway and clear the next turrets. Hack the Novice terminal to open the double-door that leads to the main lab where you can see supermutant samples in the tubes. You can pick up Brian Virgil's holotape beside the terminal to the west.
You can also find the Experimental Serum Virgil is requesting in the canister holder in the SE corner of the lab.