Sorry for the wall of show off text. There should be some good points around.
Last immortal game I built Stonehenge and Pyramids but that was all the ancient/classical wonders I got. I went wide, strong religion (lucky routes on my starting 2 scouts and got a lot of goodies including upgrades, faith, 2 techs and culture aside some stupid maps and some money). After classical, if not for that wonder spammer Incas, I would have gotten most wonders not the half I succeeded. I had no marble, no production enhancers aside the 1%prod/faithful population. After renaissance I got most wonders and in modern era, when my computer started to act up (my poor P4 and 2G DDram have their obvious limits even on tactical map), I was about equal in tech with the best civ and a clear runaway on the others.
I got most CS to be allied with me and they provided HALF my science on a wide, 8-9 cities empire - more than 300 each ( me and the CS). Rest were friendly with religion spread - Papal Primacy is a must for this.
What is to notice is that Inca, aside from getting 30-40% of the wonders started to expand their religion (and empire to about 20 cities). They even sent 1 profet and religion bombed my capital and best second city completely into their religion. I started to lose on the religion field, at least the CS closer to INCA. I recommend some spread enhancers on your religion if you go for it on Immortal or Deity. That is an absolute must.
I had 2 early rushes and I easily defended them, first with only 1 warrior and an archer (promoted from scout, those are pure gold) and the second with 2 archers and a horseman. After that I spammed 4-5 horseman to get the barbs - not too many even on ragging barbs as I played a Panagea with 40 CS because there was simply no place for them to spawn. Shall I mention I was playing Songhai and I had almost no opportunity to use my ability
? Better not.
My current game on emperor I got all but 2 wonders (G Lighthouse - no coastal early and Statue of Zeus as I was late going for that tech) on a lucky capital location with Egypt and marble - first try, no reroll. I got 65% wonder building bonus very fast with UA and religion and bought all my settlers. After my religion spread I got another 15% prod bonus and even 20% more in golden ages to a total of 100% production bonus on wonders ( I hope I did not miscalculated something but I doubt it). True, only on ancient but it was still superb. Early rush as usual, twice by Dido which ended by her losing a Turn 25 city she built next to my capital to get some elephants!, 30 squares away from her own capital! Carthage really needs those happy resources to get their elephants going? I would have gladly traded one of my 7 elephants to them. Anyway, razed and rebuilt the city as it was too close to my capital and losing some resources as well.
On a funny note, 4 of my 7 workers are freed from barbs and were Carthage ones Dido tried to ship to her far away city. Btw, Egypt's chariots are great for killing barbs. Moving 5 squares is great and ranged is wonderful. I was still killing pikes with them in medieval, job a horseman cant do so easily.
Only downside to my usual game play was the Panagea Plus map which gave me access to very few CS in early game (10 out of 30 in early medieval when I got my ships out looking for them). Not to mention that out of my 5 early CS meetings 2 were faith givers - very good, allied the asap to double my faith production - and 3! military ones which were meh.
Both games, especially Immortal one needed very careful diplomacy. Befriending everyone and not declaring wars until you are a runaway is best. Get your frindship declaration colallition and befriend but dont declare it with the ones that are the targets of everyone - A city spamming America in my imortal game and a russia capital killer Rome in the emperor one. But dont declare your hate on the singled ones. 2 reasons: 1. it is very hard to make deals unless you check diplo every turn so you wont actually have a lot to do with your spare resources in a 6 AI love fest. The singled civs always have 2-3 resources they can exhange with you and that is a good bonus. Despite being friends with their enemies, they were always friendly with me, even when I did not accept their bullying on my CS - that goes away decently fast. Second, later in the game, 2 of the singled civ haters turned to friends and I am glad I was not antagonised with my close and military advanced neighbours by an ancient age declaration. If they have nothing to offer for your resources, dont go for money - I feel it is close to stealing anyway and I dont do it unless something very important happens - but keep them around. There will always be a friend who asks nicely for them in exchange for nothing. Good diplo bonus there. And never forget that if a CS is giving you resources it never shows you the double or more amount. You can always trade the hidden surplus without an actual loss and keep around your actual double resources that will later be diplo bonuses or other opportunities.