So yesterday in my long game I went for land rights and yeah, that is the secret to either 1 city empire or just straight up mass land grabs. It converts all acquistion to Money but the curve is basically flat and not exponential like Influence.
And if you were smart by going Persians to get the city cap upgrade, you should have money everywhere with their building. I can basically attach a fourth or fifth territory per city every turn right now, verus taking 5 to 8 turns in influence (and I'm pulling in just under 800 influence in era 3 a turn on slow speed).
So if you want to expand as fast as possible, my suggestion currently is take the -50% outpost influence cost civic (normally I didn't but with land rights this changes the equation), get as many outposts as you can and attach at least one territory to every city since it is cheap. If you can get free cities from "free people" do it. Use your larger influence pool now to claim luxuries as fast as possible as well. In era 2, you should pick a money making unique building (Persia IMO is the best here) and spam it everywhere you can, and try to get a third territory on most cities depending on your influence intake to get more unique buildings (or just go market quarters). Once you get an option for land rights civic, take it and go on a spending spree with your hopefully large bank account. You literrally can triple the size of cities the turn you take it and can keep expanding them nearly every turn hopefully. You will absolutely crush everyone now since your cities will be massive.
I could probably do it faster knowing the above now, but on slow speed I pulled in 9 stars in ~15 turns for era 3 on
slow speed doing this as my population exploded, which led to more money, influence, and builder stars in quick succession. Even got the horribly overtuned expansion stars!
If you look at your empire screen you can see how to obtain era stars. The top one with the laurel around it gives bonus fame. I believe it's sometimes better to postpone entering a new era, and try finish up some current era stars quickly.
Does anybody have any advise on how to approach free states? Can I absorb them into my empire without bloodshed, for instance?
If you missed my hint earlier, try to move a scout into the city when population is zero (as long as there is no defender right next to you but you can outflank them usually by going in from the opposite side) and you get to take the city without a fight. I mainly concentrate on hostile free states since their army is usually out hunting and they don't come back after you take the city. This is harder to do on faster speeds, but on slower ones, you have a ton of time to take a city usually.