I think I'm doing just fine.
The biggest lacking factor at the moment is worker labor. All the Calendar resources, plus Wines, have my happy cap as high as 14 in some cities, so there's plenty of headroom - but the cities would be growing onto unimproved tiles. So I'm on a crash program of worker training.
Also around this time I'd racked up enough Espionage Points to get full view of the demographics, so here's a couple charts:
I think I'm doing just fine.
The Sistine Chapel is of course the #1 wonder that we need for this game plan. It took a while for anybody to get Aesthetics, but then Babylon did, so I researched up to Music and got it first. A Golden Age seemed like a fine use for the Great Artist; I'd probably get another by the time Civilized Jewelers came around. During the Golden Age, I noticed that the Pyramids were still unbuilt, and with Stone, I got them at the crazy late date of 780 AD. And with a free revolt to Representation available, I took it. I also got the Colossus northwest of the capital, after everybody else was very late to Metal Casting.
Amsterdam built the typical Great Library - National Epic combo to supply me with Great People. The first several were scientists who went for Academies. Amsterdam also built the Mausoleum, just because it looks useful.
A random event popped up:
Of course I'd like +1 research in my library, but I can't afford it. Except... that I was able to access foreign diplomacy by clicking on Examine City and then hitting F4 for the Foreign Advisor. From there I sold Babylon a cheap tech (Drama) for enough gold to pay for that event.
Yes, I still know how to scroll ahead and jump through cracks in the game's interface screens, just like the old days of Civ 3. :)
While I spent lots of uneventful time researching to Education, the game's first wars kicked off. Last-place Ragnar declaring on second-place Joao, then Hammurabi declared on Pacal, but nobody bothered me. Hammurabi then came over for an interesting conversation:
Hammurabi already captured a city, but now he comes asking me for help. Hammurabi is my best friend, at +7 relations from religion alone, and part of the ever-growing Jewish bloc. Pacal is the only Buddhist on the planet and is quite weak. I rarely do this, but I'm compelled now to join in a war at the AI's request.
Now, I could just leave the war as phony, but let's stir the pot. If I sit idle, Hammurabi is going to conquer Pacal, but I'd rather take the Mayan land for myself. So I immediately abort Printing Press research and go to war footing. The only unit option for this war is Knights; the speed is mandatory since foot units won't be able to beat Babylon to the punch. I'm three techs away from knights - Horseback Riding, Feudalism, and Guilds - so that research commences. I build a couple barracks and stables while researching the first two techs, then start building some Horse Archers timed to auto-upgrade the build orders to Knights. One Horse Archer did come out ahead of schedule, and I sold another cheap tech for money to upgrade him.
I also researched to Engineering next, for the road movement, whipped a couple more knights, and galloped towards Mayan lands.
I got there...
just in the nick of time! Hammurabi is banging down the walls of the Mayan capital already, but I'm in perfect timing - and with exactly enough force, 5 knights against 5 defenders - to poach Mutal for myself!
It helpfully came with the Buddhist shrine for 13 gold, and also the Great Lighthouse. Very nice! It also adds two clams and iron, for Sid's Sushi and Creative Constructions down the road.
So here's an updated overview shot after that:
Diplomatic chatting with Pacal shows he still has four cities left, but I can only see two of them, Mayapan and Chichen Itza. I'd really like to wipe him out to avoid lingering effects from culture and Mayan national citizens, but I can't find the other two cities. Must be out on islands somewhere.
Then, about ten minutes later, I realized the other two Mayan cities were right next to my land on the other side, Phoenician and Uxmal. 
OK, time to train more knights to take care of that.
On the research front, after that Knights beeline and Printing Press, there wasn't anything particularly urgent. With so much of my land cottaged but not yet grown, a beeline to Democracy and Emancipation seemed in order (using the Liberalism slingshot to actually pick up Democracy.)
That happened as planned, and I used the Taj Mahal Golden Age to pick a whole new slate of civics: Universal Suffrage, Free Speech, Emancipation, Free Market.
And it took quite some time, but eventually I cycled the knights back over to the left side of my empire to polish off Pacal.
