As you begin to take control, you might notice a blue glow...
Head out of the room, and down the hall, and run to cover near the windows. The prompts will tell you how to dash from cover to cover. Head downstairs, through the cafeteria, and outside, and a plane will crash, corralling you to the left. A cutscene will play, and you'll meet the first Outsiders, the Sectoids-- a race that will be haunting the human race for centuries to come. Your pistol has unlimited ammo right now, so go nuts taking them out. Head to the door to get your first supporting agent, Scout-class Nils (Rank 5).
Head through the building, eventually dropping out into a courtyard. Beyond the far gate, you'll see soldiers going by. Near them, there's a outdoor staircase, and you'll get a cutscene stranding you on the second floor. Get the M14 rife from the dead soldier near you, and open the door for Nils. When ready, open the window and fire on the gathered Sectoids. If you're good, you might be able to take them all out in 1 clip. There's a pistol down there, collect it if you want the ammo. Head out, and see two Sectoids abducting a crashed pilot. You can't do anything about it, so head around it. Watch a gunship take out two tanks (there's another rife nearby, take it for ammo).
Watch (or skip) the intro for Battle Focus, and make Nils move to flank and crit. You might be able to destroy the sentry gun on your own, if you wish. Some more Sectoids will spawn in. As Nils is fully ranked, his Bombardment can take out most, possibly all of the Sectoids if timed and positioned correctly. There's another rife and pistol lying around if you need ammo. Head inside, through, and drop out, and watch a Seed drop into the earth.
Nearing a hangar, you'll see your first Zudjari in the back: a grunt, along with two Sectoids, one near a barrel, another pacing near the elevator. For an easy kill, wait for the pacer to get near the red barrel & the other Sectoid, and shoot the barrel: the explosion will actually take out all three. Don't activate the elevator yet. But near it, or in the control area to the right, you can get the shotgun. Scattered around the room are also your first grenades. You may also want to push (just walk into) some barrels to position them: a good spot is right outside the hangar door. For when you activate the elevator, two grunts will walk in. Kill one, and a drop pod will crash in just outside. Wait a bit for the new grunts to spawn (a grenade has about the right amount of delay), then shoot the barrel to damage them and the pod. Don't forget to destroy drop pods throughout your campaign for a bit of extra XP. If you take too long to destroy the pod or the grunts, the pod may also generate some Drones.
Gather ammo, use the elevator, and head through the research lab to meet Director Faulk. After the dialogue, you'll get Engineer-class Kinney (Rank 5). Going forward, you'll see some grunts and Sectoids trying to hack open a security door on the back-right of a lab. On your end of the room, putting Nils on the right side of the door is plenty range to bombard the aliens on the right. Take care of the rest. The door will open like normal for you: head through into a tunnel. Two grunts appear at the end: a grenade is an easy kill, and there's even a grenade near an ammo satchel nearby to replenish your supplies.
Head through a few security doors and two Elerium Reactors (should you be standing that close without a radiation suit...?). After a stuttering door, your soldiers will alert you to some human soldiers under siege by Sectoids, grunts, and a turret gun on high ground to the right. After that's done: head onwards, through the tunnel (and a Seed starting to 'eat' the area around it), and down a flight of stairs-- notice an ajar (and blocked) door to your right. You can get in enough to ambush a lone (and low-health) Sectoid in the next room over. Go into the war room for a cutscene and dialogue.
Forwards, past a Seed already starting to build it's housing, and towards the rail platform, you'll find some grunts and Sectoids. Move quickly, and well-placed grenade will take care of most of them. Finish up, and get on the train to move on to a pre-rendered cutscene.
You'll come out on a nuclear bomb test platform: a low, wide loading area with many crates, a middle platform where the bomb is, and a higher helipad area. In the middle platform is the Sniper Rifle, and the radio to contact Faulk and Lt. Barnes-- doing so will activate the energy pipes in the distance to burrow around, and eventually the first wave of drop pods, so push barrels and lay Kinney's mines around the loading area to better space out the blasts, and load up your weapons and ammo. After a bit, a second wave of pods will drop. Both pod waves are just grunts and Sectoids, so no pressure. Placing Kinney's turret way in the back makes it an excellent flanking trap, and Nil's bombardment is good for taking out either set of pods.
After that, you'll have a break; the next prompt is to activate the bomb's timer, so take the time to lay another 4 mines around the bomb platform to trap the drop pods that'll come in when you trigger the bomb. Place them so (from the angle of the helipad) they're behind the crates in the walkway of the bomb platform, at the corners around the bomb. The two in the back will soften the drop pods, and the two up front will keep any Zudjari that survive from trying anything cute. The Skyranger will appear behind you, and you need to press the prompt to get in. This section is timed, but it is easy enough to kill all aliens and the drop pods before time is up (for the XP, and there is also an achievement for doing so).