Rome will be the first target, for logistical reasons, and also because he's still got only rifles without Assembly Line. I've been beelining to mechs for just this possibility (longtime readers may know my vast preference for conquering with mechs over tanks.) 
Robotics is still about 20 turns away right now, so I'll fill in the time by building battleships. (With all cities coastal on this map, battleships will serve double duty as naval superiority and city bombardment.) Battleships at a rate of one per turn, that is, with that powerful Heroic Epic capital.
After Robotics, I made one more research to Refrigeration for ship movement (Transports boosted to 9 moves with Navigation II, Magellan's, Refrigeration!), then went all cash to upgrade riflemen, galleons, and ships of the line.
I now traded to one of my vassals for Democracy, finally got rid of the Emancipation happiness penalty, and got Nationhood for more drafting. (Ouch, I didn't know mechs took 3 pop points to draft!) Anarchy skipped courtesy of Cristo Redentor.
This is annoying since I'm SO close to having Rome's vote. He shares my religion, and all he needs to do is adopt HIS OWN DAMN FAVORITE CIVIC and I'll have his vote. But he shows no signs of doing that, and since he's at war and on top of the power graph, he's got loads of police units to keep HR and he won't be switching.
Anyway, here we go.
Rome had only recently gotten Rifling, and still had many cities defended by just muskets and praets. I'd loaded and positioned six transports full of mechs. I'll cut to the chase and just mention that Rome was conquered in six turns.
Well, here's an illustration of the neat tactics possible with big ship movement. I just took Rome this turn, and by extending the war only one more turn, I can capture Rome's last two useful cities here. I have a 7-move transport in Satricum, with four mechs that took the city last turn and then loaded onto the transport inside the city with 0 moves left. This turn, that transport can sail to square #4, drop off those mechs in position to attack Ardea, sail to square #6, pick up three mechs that already attacked Rome this turn and still have a move left, and also pick up the Medic General in order to heal everybody this turn (even though they already attacked). And then next turn that transport can still reach the second square #7 in order to knock off the island city.
Anyway, after that, I accepted Caesar's capitulation. I followed up the mechs with an invasion of Sushi executives, to push my population for the UN votes.
Saladin's been on my side in voting, sharing both religion and civics (Theocracy). I noticed he lacked any "fair and forthright" diplo bonus, and handed him a gift of Medicine, and he went +4! So I can count pretty solidly on his vote.
Also, Napoleon *finally* went to Representation! I'd written off his vote as unachievable, but with a shared-civics bonus that shortly grew as large as +6, I had a good shot there. And it's time for another election, but it's a secretarial re-election.
With eight turns to go before the next victory election, I had enough time to take out another target. It would be Sitting Bull, Napoleon's former colony vassal.
He was nobody's friend (literally, not a single free leader was Pleased towards him), he's also still behind in techs (rifles as the top defender), and with only four cities would take next to no time to conquer.
Now look what I can do with the cities!
Free diplomatic points by returning the cities to the leader who originally spawned them off as a colony.
And with the other two cities, replace Sitting Bull's slot on the civs list with a new vassal of my own.
By the way, I wish I knew the exact rules for when a city can be liberated. I'd love to liberate some cities (Braga mostly, but any would do) over to Isabella, who's currently at +6 relations and needs only a little bit more to vote for me. But the option isn't available.