After conquering the Eos Vault, head to the outpost site in the Fairwinds Basin.
Drack is fighting off a few shiploads of Kett by himself. Give him a hand to trigger his recruitment.
With that done, activate the Outpost Beacon and found your first colony in Andromeda.
You have the choice to make this a science colony or a military outpost. The team will clue you in to pros and cons, but of course the final choice is yours.
Ultimately, science seems like a winner no matter what your goals. If you're looking to get in good with the locals, then it's the diplomatic choice, and even if you're lusting for conquest, everyone knows these things are won by whoever maxes out the tech tree first.
Either way, make the call and see the outpost set up before your very eyes. You'll also have brief cutscene where Addison reminds you how inadequate you are.
Afterward, you have whole new settlement to explore. Blue exclamation markers on your HUD mark the starts of the Shock Treatment and Waking Up The Future tasks. You can also browse the store, use the research station. and chat with the notable personalities around town. When talking to August Bradley, be sure to grab the Memory Trigger hovering behind him.
That's about it for the first visit, but as August says there'll be more to do here later. When you're done with Eos (either having completed your tasks or just grown tired of them), return to the Tempest for a meeting with the crew. Now is also a good time to make the rounds and advance everyone's story, and lay the groundwork for any future romances, before heading back to the Nexus.