After completing Hunter/Hunted, make your way back to Doctor Amari in Goodneighbor at the Memory Den. After talking to her about the Courser Chip, she will advise you to consult with the Railroad Faction in the Old North Church. This means you'll have to complete the first Railroad Faction quest, Road to Freedom, if you haven't done so already.
Once you've made your acquaintance with Railroad leader, Desdemona, follow her into the faction headquarters. Talk to the Railroad's resident scientist, Tinker Tom, who will be able to analyze the Courser Chip.
After Tom takes a look at the chip, fast travel to Virgil's lab in the Glowing Sea. Virgil will give you the plans to build a Signal Interceptor, which will allow you to teleport to The Institute and rescue Shaun.
To build the Signal Interceptor, you have four options:
Consult the Brotherhood of Steel
Consult the Minutemen
Consult the Railroad
Build the teleporter yourself
The benefit of getting help from the respective scientist of any of the three factions is that they'll give you advice on how to put the teleporter together. To get help from a faction, talk to the leader of your faction of choice. That would be Paladin Danse from The Brotherhood of Steel, Preston Garvey from The Minutemen, and Desdemona from the Railroad. Whichever faction you choose, you'll need to complete the first two quests pertaining to each faction, one of which being the quest where you encounter the faction. When you express your need to have a Signal Interceptor built, the faction leader will tell you to talk to the faction's tinkerer or scientist.
Whether you decide to build the teleporter with a faction or not, you first have to start with a large empty lot in a Settlement. The Minutemen's settlement at Sanctuary Hills has number of such lots. Bring up the Workshop menu and a ‘Special' category will appear. This section will show you all the necessary components.
Hopefully by now you will have scavenged a lot of building parts to fulfill all the requirements for the four sections of the teleporter. All the components use common items such as copper, steel, and rubber, but you might have to seek out three less common items: a military circuit board, a sensor module, and a biometric scanner. If you don't have these three items, don't worry. The Pip-Boy is programmed to point out the likely locations of these items.
The faction scientist will help you after you've built the first section, the Stabilized Reflector Platform. Then build the other three components: the Beam Emitter, the Relay Dish, and the Console. Note that these all need power, so you'll have to build a number of generators. Note that the Beam Emitter needs at least two generators since it requires 20 units of power and the most powerful generator you can build can only generate 10 units.
With all the components and generators built, now is the time to properly assemble the teleporter. There are two key tasks. The first is to make sure you've attached wires from the generators to the four sections of the teleporter (you can see the wire button prompt when highlighting a generator in Workshop mode). Then you have to make sure that the Beam Emitter is directly on top of the Stabilized Reflector Platform. With those tasks complete, you'll be able to activate the teleporter and complete this quest.