Azzy is the least played civ in the game, which is undeserved. The reasons are because of how misunderstood they are, their unique style and possibly their strength. They are one of the top civs after all. They are also a TWC civ which means their villagers have that time-wasting, awkward animation when firing at hunts. This is one of those things that can make you dislike a civ without realising it.
Being a native civ, Azzy also lack market upgrades. The wood upgrade is decent but expensive, and not having steel traps is a huge deal. Really, don't underestimate how big of a deal this is. With 16 vills on food which is actually pretty common in mid colonial, steel traps would be worth 4 vills and it only costs 250 resources.
Pumas are cost effective units but they cost 50f/50g. They are the only Aztec unit in colonial that costs gold, and it's hard to plan how many pikes you need. Usually you'd want the option to simply mix them in on the fly, but it doesn't quite work that way with Azzy. This can make managing your early economy somewhat awkward, but it's something you get used to.
Azzy's early game is complex. There are a lot of choices concerning TPs, and a lot of different shipment orders as Azzy gets an insane amount of shipments early on. They have the option to rush, and this can be viable in some match ups, but the end result according to Azzy's game plan is always a prolonged colonial war as that is where the civ excels.
They ultimately play for the late colonial. They scale extremely well in colonial due to their strong composition (have you tried to micro against mace+puma+coyote?), the war dance, and the excellent coyote upgrade card. However, their boom is limited and some civs can end up outmassing them and can break through their (forward) position in mid colonial, or possibly even age up. In most cases though, Azzy will hold onto map control or hit a strong timing against (semi-)FF builds.
It needs to be mentioned that 3 WP is one of the best colonial shipments in the game. It equals about 2.5 TPs in XP income (3 XP/s), and you can switch to war dance for extra stats at any time (which is a pretty big deal) or even spawn more WP if you're planning for the long term. This is a very strong shipment to send in early colonial because it speeds up all your next shipments by a lot, and gives you the option to buff your units significantly in fights. But is it a better eco shipment than 5 vills?
Yes. Yes it absolutely is. Not even close.
The Aztec build
Immediately check the amount of wood crates you started with. If 3, you're going to want to build a TP immediately (unless you're rushing, then go for market). Get the fire pit up too, and chop 100w for a house. On GP where you can take the middle TP, you'll need 4 vills on wood to get this on time, as your first shipment will be ready at 1 minute and you'll be on 12/10 pop. This will slow your age up slightly and you may even have some idle TC time early, but it's absolutely worth it. If you started with 200w, simply go for the house+fire pit.
Fire pit goes on XP dance until you get your first shipment, then baby dance until you start ageing up. This allows you to make an extra vill but not age up that much later.
Azzy's politicians are pretty terrible. Thank god for the fast age, which is the best politician in the game and should generally (with all civs) be used whenever possible, unless you're saving it for fortress in case of an Iro semi-FF. Azzy can semi-FF, but they don't care as much about saving the fast age because they don't mind building colonial units on their way to fortress, something Iro is not a fan of. Anyway, why is fast age so strong?
The minute faster age up gets you +3 vills due to less idle TC time, and since Azzy has 2 or even 3 shipments ready as they hit colonial, ageing up slowly would be extra painful because your first 3 colonial shipments will be a minute later than usual.
When you start ageing up, leave 4-5 vills on food for a steady vill production and put the rest on wood. A choice you have to make here is whether to chop for a TP first, or go for the WH immediately. This depends on your game plan, if you go for a TP first you may have trouble holding off rushes and you won't be able to put on early aggression other than small coyote raids, but it is the fastest boom you have so if you're playing for the long term, this is worth it. Azzy is a civ that is very good at playing for the long term, and this plays into that.
As you enter colonial age, you should have almost enough wood for either the TP or the WH. I'll leave it up to your judgement when to build the market, the most greedy eco play is to delay the WH until 700w and chop for a TP+market, but the order at which you build these buildings is up to you and depends entirely on the situation.
(2) 700w (exception: no eco rush)
The rush 10%
Azzy have a no eco rush where they chop for a WH with all their vills (meaning idle TC time in early colonial), get it up asap and forward and send a unit shipment first in colonial (so no 700w) which will arrive at that WH. This can be strong sometimes, after all having 10 maces in your base at 4.30 is really quite annoying, but minutemen shut it down and it sacrifices Azzy's own eco as well which ultimately makes it not worth it. It can be a decent meta choice against civs that are unlikely to have the resources for minutemen, though.
The more well-rounded and more popular rush build plays for strong pressure around 6 to 8 minutes and does send the 700w first. Note that if you were planning to rush, you shouldn't have went for either of the early TPs as they will slow you down a bit which in case of a rush is a big deal. If you don't have a market yet, build it from the 700w. It's an option to build a TP from the 700w. This will result in a slightly weaker rush but obviously a better eco, and is often worth it if your rush is just meant to force them to stay colonial, which is about the only situation in which you should be doing this rush.
(3) 10 maces
Note that it's better to mine for the pumas than to send them. Push with mace+puma or mace+coyote depending on what you made from WH, and decide whether to continue the pressure. As long as you're continuing the aggression, keep sending military shipments (9 mace and 6 puma, and even 5 coyote (which is a bad shipment) if you really need the units asap). As soon as you think it's not worth it to continue (his defense is too strong), or you've accomplished your goal in that he was forced to stay colonial where he would otherwise have aged, it's time to set yourself up for the long term and send 3 WP. Sometimes 600w first is necessary to keep up at least some military production, depending on if you need that or not.
The 8 minute timing 30%
Meant to punish semi-FFs but also viable to punish heavy eco play, this is a timing that hits hard. It damages the enemy's economy and sets Azzy up for a prolonged colonial (often vs fortress) war where Azzy's primary goal is keeping map control.
When going for this timing, the second early TP is somewhat awkward. It means your mass at 8 min will be slightly weaker, however it will get you the 9 mace shipment earlier which means you can hit your timing slightly earlier. More often than not, you should build it from the 700w. Remember it's also very beneficial when it comes to the follow up, you will need a strong eco in the end. Generally you'll have 3 more shipments after the 700w before you push:
(4) 10 maces
(5) 9 maces
Note that you don't need that second WH just yet, you will be able to spend your resources just fine without it.
Sometimes it's worth it to keep up pressure with 6 puma, but in most cases after the timing you'll want to invest in economy. 3 WP is the most common follow up, allowing you to crank out 5v and the coyote upgrade quickly, still having 600g for situations where you need a lot of puma. Civs who went for a semi-FF and faced this timing will have a lot of trouble breaking your position, even with falcs.
The late game plan 60%
This is the style that Azzy secretly wants to go for every single game, because it sets them up for a solid mid to late game where they are stronger than most civs. But they run into the issue that if they boom, their opponent may age up and even though Azzy can fight fortress armies in colonial and can even sometimes age up themselves, allowing the opponent to comfortably age up does create chances for them. In such cases, think France for example, Azzy can still start eco-heavy but may adapt into a somewhat weakened 8 min timing with a stronger eco behind it, if they scout a semi-FF from France.
The most greedy start Azzy can do is delaying the WH until 700w, chopping for a market and TP, and sending the following shipments:
(3) 3 WP
(5) 600w (second WH) or 600g (age up)
Needless to say this will get you a really strong economy with either 1 or 2 TPs, and your dual WH mass will soon overwhelm your opponent. However, some civs will abuse your early passive play and age up or possibly even push in colonial, in which case you can opt out of 5v and send military shipments. Against (semi-)FF builds you can also choose to age up after them with 600g instead of 600w. Note that 5v is not important, it's just stronger long term and if you find yourself needing units for a timing or to hold a timing in (relatively) early colonial, skip 5v and send maces.
Against early colonial rushes, this build is vulnerable so make sure you scout your opponent and make sure he isn't pulling any of that stuff. If you scout a fast rush simply send 10 mace instead of 3 WP and delay your boom a bit. You should still hold any rush that way.
If you ever find yourself in fortress (age up with the nobles hut politician), keep going mace+coyote but add ERK instead of pikes. ERK are an extremely strong dragoon-type unit and will demolish any cav he throws at you while still providing some decent damage against infantry too. Against falcs, make some arrow knights. These are good for sieging a base from far away as well.