My general style is rush until close to 0% science, rebuild till about 60-70% science, rush, rebuild, etc...
I realize that this may sound like blah blah blah rhetoric, but play what the game gives you. Don't pigeon-hole yourself into an early rush.
I have found in many situations that an early rush works poorly unless it is handed to you on a silver platter. What I mean by this is if you get Bronze out of a tribal hut and it is in your bfc or if you have horses in your bfc, give it a try. I found in my games, I was unsuccessful in an early rush if horses or bronze was in the third ring of culture outside the BFC. That being said, it may be doable if you are a creative civ and can pop that second culture expansion earlier (I really haven't played out that scenario so I don't know). If your nearest neighbor is a Protective civ, forget about it. If he is not protective, but he built on a hill, good luck.
I found that a key to my immortal games is to prioritize basic worker techs, mining, Agr, BW, AH, +/- fishing, or less likely hunting. Then go for writing. Next try to build/chop a settler and a library. Run scientists to help your shrinking economy.
My secondary tech focus is on CoL, Aesth, or Alph. All are great trade bait. If you go for Aesth or Col, you may have to research 1 or 2 turns toward Alph at 100% on the slider since the AI is a real trade tight-wad on Immortal.
I rarely have problems with getting dog-piled early on unless you adopt a religion. Get open borders ASAP for the diplo boost that will follow. If all civs are Budd, Hind or Jud, then take the religion.
Worker stealing can be a huge benefit. Rarely, on a normal speed game, will I have an AI that will actually bring troops after me after an early worker steal.
Barbs can be an issue. I have been pretty successful with fog-busting with warriors. The AI will not spawn within two plots of your units or borders. So while I am growing my capitol, I build warriors and station them in a wilderness perimeter around my capitol, ideally on forest/jungle hills, or non-hilled if none available. These will limit spawned barbs and will take care of most barb archers and almost all warriors. (Sometimes they will slip by the fortified units, then improvise.) If you get bronze or horses in your 1st or 2nd city bfc, this obviously helps with barbs. Obviously, archers are better for barbs, but I can get away with delaying archers.
If you're stuck on an early rush, I would suggest bee-lining Horseback Riding for a horse archer rush (or Numidian or Keshik). (Don't forget archery.) My horse archer rushes usually have a few axemen and end with a cat or 2, since I usually get construction near the end of my rush.
I feel a GL/NE sprint while Rexing subsequently bulbing and targeting Lib helps to spring me to sit that opens up options by putting me in a tech lead. I don't waste time with wonders either. Although if you are playing a map with a lot of open land or raging barbs, the Great Wall is a must. You also don't need Stone to grab the GW.
I bail out my dying economy by getting the library built early and running Great Scientists, and then trading Col, and bulbing/trading Philosophy (and Paper if I have to.)
Having gold or gems and lots of food in you starting BFC are also a game breaker.
Since then I've improved and can conquer a neighbour with Horse Archers, but the key realization was that Axeman rushes are no longer automatically guaranteed.
