Standard Poke
Typically, this is done by W->AA, W->Q->AA, or W->AA->Q->AA, in increasing order of damage dealt. Remember that Q slows the target, so landing that can sometimes even result in two attacks at the end.
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Standard Dash
Just press E on a soldier to dash to them. If you do not press directly on a soldier, the game will try to find a soldier near the mouse. Sometimes it isn't the most accurate, so make sure you are.

2-Soldier E
Just because you don't hit someone with your E doesn't mean that you still can't ult them backwards. If there were two enemies, you could knock up the one in the rear before ulting both back toward your turret. Don't always go for the nearest enemy. Be aware that when you have two soldiers out, neither one will end up directly where you're aiming. You can also use this to your advantage when trying to flee and you're afraid of being body-blocked.
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Long Dashes
If you want to have a longer dash, you can actually press Q while you're in transit on your E and you will follow the soldier to the new destination. Now, the obvious implication is that you can dash farther away without having to wait for Q to finish casting like you did before, but there's more than that. By timing this right, you can bait out Flashes and other blinks by going over walls in ways your enemies won't expect. Be warned that timing this can be difficult and I myself only use it when I would die from the conventional method.
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Sweep Farming
The fastest way to kill a large wave of minions bearing down on your turret is to place a soldier beside the wave, E to it, and then Q across the width of the wave while you are in transit. The damage from your Q and E as you sweep across the minion wave will be sufficient to kill everything you hit. Drop another soldier and kill any stragglers that you missed.

Blue Buff
As obvious as it may seem, Blue Buff is incredibly useful on Azir. Get it as early as possible. It's not every game that you'll be able to get Morellonomicon and Nashor's Tooth super early because of defensive itemization against Zed for instance. Sometimes you'll need some CDR to hold you over until you do.

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Soldier Ward
In many cases, you won't have a ward ready from your trinket, but you'd still rather not die. In such cases, you can simply drop a soldier over a wall and let them act as a short-ranged ward.

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Turret Backup
When you're advancing during a siege, few things can say "don't counter-engage" like summoning a turret on the enemy's ruin. If they attack, all your team has to do is walk back a couple hundred units and they have to face the wrath of an angry tower

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Rounding Up
Just like dropping a ward just over the half-way point of a wall, if you drop a soldier with W or send a soldier with Q to just a bit past the mid-way point of the wall, they will end up on the other side.
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Push-Proof
Sometimes you really want to leave lane, but don't want to lose CS or lose lane pressure. In such situations, you can set up your turret and leave it be. It'll give you the gold from the minions and you're free to go off to see the wizard or whatever else you might wish to do while your turret babysits the lane.
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Last Line of Defense
If the enemy is attacking your nexus and getting greedy by attacking people on the fountain, don't be afraid to W->Q->E->R them onto your fountain. Sure, you'll probably still lose the game, but at least it'll serve them right for being so greedy.

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Escape Hatch
When you're pushing down an enemy turret, it can be tempting to just drop both of your charges immediately to drop its health as quickly as you can, but if you don't have good vision, that can be a bit suicidal. Leave a soldier in the back that you can quickly Q in a safe direction and then E to should you get flanked.
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Impromptu Cage
Azir doesn't have any long-duration CC, just several short little bits of movement disruption. This can sometimes make it difficult to really get all your damage on the target. After all, a moving target is much harder to hit with a stationary soldier. To fix that, find a concave wall and simply ult you target into it. With proper angling, it can even be used on turret ruins. Once they're trapped, just Q your soldiers over and have fun turning them into Swiss cheese.

Everything I've mentioned up till now in this section has been pretty straight-forward. "Do X, get reward Y." What you'll see here are plays that almost always result in your death or at minimum result in very heavy damage taken. You only want to do these high risk, high reward plays if you can minimize the risk or your team desperately needs the reward. Sometimes you'll be forced into plays like this though and you have no choice but to take a stand. Pull these off right though, and you'll see chat explode and will sometimes trigger an immediate surrender.

These are the sorts of plays that when done right show people what the skill cap for Azir is, something that you almost never see. It's easy enough to try all of these. Situations for them arise in nearly every game. The trick is recognizing when you can and can't capitalize on them.

Dragon and Baron Steals
The easiest way to steal Dragon and Baron Nashor is to start behind the pit wall and drop a solder inside the pit, Q them to the other side of the epic monster, E to follow your soldier into the pit, and then ult toward the mouth of the pit. This will shove out most people from the pit, particularly melees, meaning that they can't smite it anymore. From there, just use your soldiers to cut down its health while the enemy team watches in horror. You'll probably die if you don't flash out though, so be warned. Obviously having your team nearby helps with this, but I have done it solo many times and those plays have turned around games we would have otherwise lost.

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Long Range Engages
Azir can engage on enemies from a surprisingly long range, to the point that the enemies didn't even have him on their screens. The absolute maximum is Flash (425) + Arise! (450) + Conquering Sands (875) + Emperor's Divide when used sideways (300) = 2050. For reference, Flash+ Summon: Tibbers is 1025.
Unstoppable Force is 1000.
Solar Flare is 1200.
Arcanopulse goes to 1400.
Vel'Koz's ult is 1550.
Piercing Arrow goes to 1625.
Azir's maximum engagement range is 2050. I recently actually had the fortune of one of Azir's long-range engages. We were winning the game, but the enemy team was still at a point where if they got a solid win in a teamfight, they'd win. We had just pushed them off Baron, so they were fleeing back to their base, though with minimal damage taken. In the image below, you can see where I started and the ping is where I ended up. I dropped a soldier over the wall, then flashed over it, then Q'd to the enemy team, then E'd, knocking up their ADC before ulting them all into the wall. I caught four of them in that ult that pinned them to the wall.

Yes, I died in the ensuing fight, but I was the only one on our team and we aced them. As exciting as this play was to make, it's not even as long as it could have been. As you can see in the image, it's nowhere near as long as it could have been. Even so, such a surprise attack caught them completely off-guard and we induced a surrender.

These sorts of attacks should be weighed carefully. I knew I would die going in. I was at less than full health and I was about to be in the middle of five very scared enemies with the nearest ally about 800 units away at first. Even so, I knew our ADC was close enough that he'd make it in time for the fight and could clean them up after I'd taken out most of their health prior to dying.

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Ulting Over Walls
Just like anything else in the game, if a champion gets more than halfway through a wall due to dislocation CC, they'll end up on the other side. Put more plainly, Azir's ult can knock people and monsters over thin walls. This is the safest thing in this list as you only need to be in range to ult them, but when done properly, you can permanently cut off half the enemy team from the fight. Imagine a fight in the dragon pit and suddenly you W->Q->E in from the mouth of the pit and knock two enemies back over the pit wall. They're completely out of the fight now and there's nothing they can do to get back if they don't have flash. In terms of securing objectives, this is effectively as good as outright killing them because it prevents them from contributing at all.
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Fighting Under Your Turret

Fighting under your Sun Disk is a lot like fighting under any other allied turret, but with two major differences. First, everything it kills counts for you, so you don't even have to mark an enemy to get a kill from it. Second, and more importantly, if you're at a normal turret, you're probably at least vaguely close to another allied turret you can fall back to if things go particularly sideways. When fighting under a Sun Disk on the enemy side of the map, there is nothing to fall back to. You either win the fight or you die. There is no running from it.

If you want to win these fights (specifically, 3v1s and 4v1s), you need to be at least semi-fed, you need to be full health, and you need to have all your spells (including ultimate and summoners) off cooldown. I have been in three fights such as this and all of them were successful, but only because I had access to every tool in my kit. As I recall, I had use Zhonya's Hourglass in the 4v1 fight as well. If anything would have been on cooldown, I would have died.

The first thing you should do if a fight is imminent is hide just behind your Sun Disk so that as you move around in the fight you stay within its radius. Drop a soldier beside it. If there is incoming dislocation CC like a Blitzcrank or Thresh grab, you need to dodge it. If you are hit, you will die. As the enemy melee champions charge toward you (remember, we're talking cases where you're vastly outnumbered), move toward the center of the lane. This will curve them off a bit, pushing them toward the wall near the Sun Disk. The moment you take damage (or before then if there are no minions around), ult them into the wall beside the Sun Disk. This will render them trapped and stuck with turret aggro right beside it where they will be hit almost instantly by its shots.

Attack them with auto-attacks while the ranged champions close in. You must get the kill. Target the greatest threat that is in their back line by WQE stunning them as soon as they fire on you to draw the fire of the turret from the recently deceased. Immediately flash back and run to your now decomposing wall. Use the speed boost from it to kite as best you can while stabbing away at their back line as they attempt to tower dive you without their front line. Ignite whoever is closest and focus on whoever is the greatest threat using standard attack combos. Your Q slows them. Use that to keep distance between you and them as you slowly back away, but never leave the turret's attack radius unless it's for a brief period and an enemy champion does still have turret aggro.

In surprisingly short order, almost before you realize what has happened, the fight is over. Three enemies are dead and one is scrambling and Flashing away as quickly as they can for fear of the turret that has a red laser pointer aimed at their head and as their health bar ticks down from the last blast that left them with only a couple hundred health. You survive with only a couple hundred health remaining and limp back toward your base for a while before feeling safe enough to recall.

These fights are incredibly dangerous and risky. Pull one off though, and you might just trigger a surrender right there (one of the three did result in an immediate surrender). The trick is to narrow down the fight so that while it is a 4v1, at any given moment, you're just doing a 1v1 with the help of your turret at best or a 2v1 at worst. This is not the sort of thing a new Azir player is going to be able to pull off. In a 4v1, you have to be running on instinct, or you will die because there is no time to think.