This multipart quest weaves through the length of the game and the entirety of the crew- culminating in a Wicked Gracelike bit of fun for all the characters.
Chat with Liam onboard the Tempest, and he'll eventually ask you to download a movie database from the Nexus, so he can throw a little cinema party.
Of course, SAM could do this in a trillionth of a second, but you prefer the personal touch. Next time you're in the Nexus, stop by Operations to manually download the database in question.
Talk to Liam aboard the Tempest again to close out this section of the quest.
You'll get an email from Suvi about movie night.
Talk to her on the bridge and she'll tell you that she wants to bring snacks, but like any good NPC she can only make do with rare, alien, snacks that you have to procure personally. Luckily there's no dragonslaying involved, just visit the General Goods Merchant on Aya (after the Angara grant you access to city, of course, but you probably can't even get this part of the quest until they do).
Buy the Tavarl plant, in the Special Items section of your inventory (the book with the gear icon).
Report back to Suvi to end this part of the quest.
The next part starts with an email from Vetra, though it may simply trigger in conversation as well. She's not a fan of Suvi's cooking, and has a better alternative- contraband popcorn.
She's located a source on Kadara, so drop by when you get a moment. Purchase from the vendor (with your own hard-earned credits), and return to Vetra to close out this loop.
The Right Setup
After successfully hunting the Archon, you'll get an email from Jaal with concerns.
Talk to him aboard ship to learn about Angaran moviewatching habits. He needs some gear to make the experience complete, which of course means he needs you to fetch something for him.
Get the supplies from the Aya Market and return to Jaal.
Raise a Glass
After completing The Journey to Meridian, talk to Lexi to hear her concerns about home-made alcohol. She'll send you on a booze run to the Nexus, where you can pick up some Akantha from the general goods merchant.
Return to Lexi to complete this segment.
The Final Piece
Now talk to Liam, who'll send you on a quest for the perfect movie.
Return to the mods merchant on Kadara to acquire this masterpiece.
Return to Liam and he'll queue things up. Now you just have to return to your quarters and activate the beacon to get this party started.
Let the cutscene roll, and have fun with it (note: there is a bug where Ryder becomes invisible during the very last shot. Just try to have fun with that too).
Completing this quest automatically gives you a model of the Tempest for your wall.