My typical Aztec start opens by building another Jaguar for scouting, and killing the first Barbarian I meet. That gets me to my first social policy, which is the Honor opener for more barbarian hunting. After that, all social policies are Tradition and are written in the blood of Barbarians & anyone who pisses me off. I often round out Honor too because war is important to me, and getting gold + culture from kills is good fun.
First tech is Pottery, typically to Calendar, and then to the Wheel for the Floating Gardens. I generally go with Calendar to improve luxuries nearby so I can sustain the growth the Floating Gardens provides. (I might have to go off-track for Mining if I have those luxuries, or Bronze Working to clear for them).
If possible, I'll nab Stonehenge or the Hanging Gardens, but don't shed too many tears over either.
I'll generally go for a 4-city build to start, and expand as I see fit later. This can either be by going to war with the neighbors to make some puppets, or by settling a bit wider. Tenochtitlan isn't usually coastal so I don't sweat getting a second coastal city early to ship food around.
You don't always have to be at war; the UA has done its job if it catapults you faster up Tradition. That said, being at war IS often good for you provided the enemy can keep feeding you tasty packs of culture on legs. I often find myself fighting like the historical Aztecs; more for the glory & fun than really conquering much. If you're careful you can keep decent diplomatic relationships simply by accepting cities as tribute instead of capturing. I'm also unafraid to burn swaths of land out & let it fill in with tasty Barbarians who are worth more gold & double Culture.
My army tends to run a little infantry heavy. Super-promoted Jaguars can be hilarious, with March, full Cover, and full Medic they're pretty much unstoppable 1-A across the map units.