And here we are. The barbarians are gone, we're in Republic, and ready to rock.
I need a Forbidden Palace. I'd been planning to build a city somewhere around the Houston area to groom specifically for the FP, but then America moved in with their city. The only other option was the powerful city of Allegheny. I didn't want to lose my excellent worker farm (once I could plant some forests on the wine tiles, it could make a worker every turn), but the FP was more important. The city starts on it, and with the cattle and wines, rockets up to size 12 very quickly and finishes the FP in 20 turns total.
Also 20 turns later, the Republic payout expired and left me with plenty of income to spend. I bought all the techs that enabled buildings: Mathematics, Iron Working, Currency, Construction, and Map Making, for a total of about 70 gold/turn. Then after my first couple of marketplaces completed, in 670 BC, I spent another 30 gpt to pick up The Wheel, Horseback Riding, Mysticism, and Polytheism, and enter the Middle Ages.
All those tech purchases had been from America or Zulu, not Babylon. I'd noticed early on that Babylon was a ways behind on the power graph and only had few cities. So I wanted to keep Hammurabi behind, in order to lead up to this happy day:
I'm in the brokering loop, baby.
I had to guess that Babylon lacked Invention from the fact that they weren't building Leonardo's while everybody else was. I gambled on buying Feudalism now at full price, and the guess turned out to be on the money: I could buy Invention and broker it to Babylon for Theology.
Soon after, I also two-ferred Gunpowder for Education and Chemistry for Banking, both with Babylon.
And now that I was just about caught up, I decided to go for a big payoff with a wonder in Allegheny, at 25 shields/turn. I was hoping for the Sistine Chapel, but England beat me to that by 7 turns. Then Sun Tzu's and Leonardo's both fell, but somebody researched Music Theory for me just in time and I even got to broker it for Astronomy. Bach's was complete and that let me drop luxury tax to 0 at last.
The other cities were pursuing typical builder plans: aqueducts where necessary, granaries, marketplaces, cheap temples and cathedrals, courthouses, even some libraries. Niagara finished all that stuff pretty quickly and started a wonder prebuild as well. I parked a scientist on Printing Press as usual.
I also bought Physics to broker for Navigation plus finally everybody's world maps. I had built a good half-dozen galleys in ancient days in a bid to make contact with the two missing civs then. A couple of them tried early suicide runs, but soon I gave up on that and had the rest of the boats wait in good positions to be ready to run once ocean navigation came around.
The Zulu had had Navigation for a good 20 turns by now, but they still hadn't yet found the missing civs. So I leaped into the race for contact myself... and bingo.
England has contact with Egypt, and I trade an outdated tech for the contact, all the remaining optional techs I didn't have, Monarchy, Chivalry, and both civs' maps and cash.
Total tech parity in 130 AD. Ding dong, the ditch is dead. 
