So I finally played this out.. Was real fun! I'm not gonna do a big write-up, as that would be of limited value as I'm still pretty new at this level.
I did take some advice from here.. I slow-built and whipped out some infrastructure to get a better economic base for warfare. I then took out Gilgamesh (nearest neighbor) in two instances -- first capturing his closest cities and his cap, letting him re-settle some spots in the 10-15 turns of peace while I re-grouped and re-built some units. Took him out almost exclusively with Numidian/UU).
After/during war I sailed around the globe, met the other continent. I was backwards in tech, but got up to speed by trading and stealing techs from Darius. After I took all his techs I declared and took him out quite fast (upgraded UU to Cav).
From there and in I was solidly in front, and I could pretty much pick and choose victory. I went for space (around 1940.. not very impressive, but first time at Monarch, so I'm happy enough) while taking out the one civ on the other continent that was not in war.. just for fun, really. The war lasted 3 turns (modern armor vs rifles.. mostly).
Could probably have won by Diplo some time before 1900, didn't cause it seemed kinda boring

, or domination around the same time as the space victory.
I have a lot of saves if there's any interest.
Lessons learned:
Espy was fun -- was the first time I've ever used it. I switched between 100% espy, 100% science and 100% gold, and got up to speed on techs real fast, stealing techs every ~6-7 turns. Religion modifiers and stationary spy made it very cheap compared to self-teching.
Played the other continent quite well -- was only intermittently able to get the largest civs there to friendly, but was at all times the "best liked" civ on the map. Gave in to some demands and ran a horrible deficit on trades, and was never worse than pleased. Monty and Peter was cannon fodder on the other continent, and I planned on taking out Monty to get the war struggle diplo bonus, but was too late as Monty vassaled to Babylon. Peter vassaled to Sitting Bull, and the two vassal powers ended up in war, predictably enough.
All in all very enjoyable game. Thanks again for the tips! I'm tempted to try Immortal next...
