The Eos Vault
When you enter the vault, you'll have a chance to alter your loadout. This will be a pretty combat-heavy mission, so make sure you're using your best weapons and mods, then head on in. A cutscene will show the group encountering an empty room, then finding a nigh-bottomless shaft leading to mystery. You have a quicktime event to leap down the shaft before Peebee does.
At the bottom, scan the spindly Remnant artifacts. You've encountered at least one before, but these appear to be infinitely scannable.
There are two Containers in the room. If you're standing at the Power Console, one is down the ramp to your left, the other is in the wreckage behind you.
Loot them and scan the door before activating the console. In theory you now need to follow the power conduit with your scanner, but between the linear level design and encouraging quest markers, you hardly need bother. Press ahead, scanning everything until you reach a room with a giant energy column, and Peebee splits from the party.
Do not attempt to enter the column yourself- the electricity will take out life support instantly. Instead bear left/west and up a ramp to note a door with a barrier. You'll be back for this later.
Follow your marker to the next room. On the southwest corner of the platform with the Remnant Console is another Container.
Leap across the lake using the pillars. You can scan the giant mechanical arms in the middle, and another spindly device just past that. Engage and destroy the Remnant force at the far shore (starts out at about 5 Assemblers, but of course it doesn't stay that way for long). Then search the high corners of the room for hidden loot.
There's also a locked door up there, but leave it be. Go to the precipice with the view of the city. While your marker bids you go south, instead take the north path, and enter a small quadrangular room for a Container.
Remember this configuration of ramp, room, and container, because you'll see it repeated a number of times. Now follow your marker, You'll pass another quadroom like the one you just looted- loot this one too. Onward you'll encounter another handful of Assemblers in a multi-level area. I don't mean to keep going on about great the biotic charge is at taking out dispersed enemies like this, but it really is a pleasure. When done, snag the container in a upper south west corner.
Follow your marker for a cross-chasm convo with Peebee. Don't immediately enter the big door, but instead round the far corner for a container in the rubble to your left, and also through the door on your right.
Now you can go through the big door. Scan the Remnant Lights here (you'll see a lot more, but you can only scan one), then follow the path into another gravity well base. There are five Assemblers and one Observer, but there's also an auto-turret up on the north side. Activate it for a little friendly Remnant fire.
Smash the other bots, and remember to scan the turret itself before taking another dive down the rabbit hole (by activating the other console, then the well in the middle of the room.)
There are containers on the south side of this chamber, and on the north platform by the light fixture. This north one contains a Remant Cryo Gauntlet, which is pretty great, and can be equipped immediately.
Scan the console to confirm that it's broken, and Peebee will shout down some advice. Scan the leak behind the console to implement it.
Activate the console, then out the door to a more arboreal chamber.
There's another spindly thing to your left- scan it then activate the console to grow some jumping pillars. Hope across and scan the central tree.
Activate the console and hop to the next platform (the one toward your left). There are some baddies waiting for you- Assemblers and Observers both. Use cover to hide from the latter while taking out the former, then mop up. When you're done, there's a larger door at the very end of this platform. Go through, and Ryder will mention it being a side room. Further on is a hall full of inactive Assemblers.
Despite some scares, you can loot the container here withough actual repurcussion. Do so, then exit back through the main door, and activate the Remant console. Now you'll have to backtrack to the last platform you hopped in from, but as you do another three Observers will strike.
Hop back, take the right branch at the last platform, and follow the new path. The last platform will have a mix of Observers and Assemblers. Hop over to defeat them, or charge instantly over, or snipe them from where you are.
Clear them, then use the console to spawn another flash bridge. Before entering the door, explore the upper platform here and bear left for a Container.
Enter the door into a fountain room, and grab the container against the northeast side.
You can't scan the trees or the machinery or the energy stream, so just activate the Remnant Console when ready.
This triggers a cutscene, and also a giant deathcloud. The next sequence is a sprint for the exit, with minor hazards. The first is that the big square vents in the floor belch fire, so don't run over them.
Past the vent you'll need to activate a console for a flashbridge, then just follow the marker up some ramps, through a door, and up a gravity well. This leads to the room with the barrier from earlier- quickly jump up and loot the container inside.
Sprint for the exit to trigger a cutscene, and a revelation.
When you surface, Peebee will join the party, and you should follow SAM's marker to found your first colony.