An obvious place to start this bout of exploration is on Aya, of course, which you can now explore more or less fully. You have to trek through the place again to get back to the Tempest anyways. Starting in the Resistance Headquarters, talk to Evfra (and hence, the Moshae) for more information, if you wish, then scan some displays to the west and read a Communications Terminal to the east.
Other than that you can check out the infirmary and training grounds (both downstairs). In the former you can learn about Angaran physiology by talking to Olvek (after you use the obligatory Decontamination Console, of course) while in the latter you can listen to Cora babble about Asari huntress training for the hundredth time, as well as talk to Quartermaster Istaal, who will begrudgingly sell you weapons and mods.
Repository of History
Leave the Resistance Headquarters and from there head northwest to find an Angaran named Daanfre, who scanned you on your way past earlier. Ignore her a little bit longer - you’ll get to questing this place shortly - but first continue east from the overly-inquisitive Angaran and you’ll reach a door that seals off a section of the city. Ignore it for now and head north to reach the Repository of History, where you’ll find Peebee poring over the Remnant Museum Exhibit. Talk to her, examine the exhibits, then talk to the Angaran staff - Nalon, Elihn and Avela Kjar. Sadly, they seem to know relatively little about ancient Angaran history and the Scourge, which caused its downfall and a subsequent dark age, and it’s safe to say you know far more about the Remnant than they do.
Exit the Repository of History building and head through the door to the west to find a bar… or restaurant. Or both. Either way, scan some Nutrient Paste Vats behind the counter for some Research Data points, then note that both Drack and Liam are wasting time here. Talk to the bartender Roaan Aabel to learn about the Angaran food processing techniques and possibly sample some Angara beverages… awkwardly. Isn’t it odd how they seem to use the same, plain metal cups as the Nexus does? Strange.
Head south, then west up some stairs to reach the Aya Marketplace. Interject into a forum discussion, then continue west to reach the business end of the market, where a variety of merchants are peddling their wares. This includes a Materials merchant selling augmentations and materials, including rare resources like Vanadium, Titanium, Platinum and Element Zero. A General Store Merchant sells consumables and more materials, Felaan peddles raw fruit (which is of no use to you right now), while another merchant - Xaan Ingil - is less friendly. Finally, if you talk to Arbiter Renaav you’ll learn a bit about Angaran laws and trade customs, starting the quest "Task: Laws and Customs".
From the Marketplace head west, then turn north and head upstairs to reach the Governor’s Office. In the front room you’ll find your old pal Enroh Bosaan, who is eager to impose upon you the quest "Task: Messages to the Nexus". Talking to the adjacent Arbiter Ohmdaal will result in him giving you "Arbiter Ohmdaal’s Datapad", in connection with the quest "Task: Laws and Customs". Ironically, the only person here who won’t pester you with quests is the governor herself, Paaran Shie. She does have quite a bit of chatter you can sift through, however.
Leave the Governor’s Office, return to the Marketplace, and from Arbiter Renaav head west, then south to return to the docks. Along the way you’ll find a Display Stand for a memorial garden and an Angaran Mural you can scan. If you go through a door to the southwest of this mural you’ll reach the control center for Aya’s docks. Grab an Angaran Shuttle Model off a desk for your toy collection, note the handy Research Center for your crafting needs, then talk to Maariko. As local astronomer, he knows more about the local systems than most Angara… but recently some tools of the trade - several monitoring satellites - have gone missing. You have a ship, he has a need, and together, this starts the quest "Safe Journeys".
With that, you’re done exploring Aya, but there are still plenty of quests to do, which will be covered in the next few pages, and after that, some follow-up quests in the Nexus will be covered.