I haven't had any luck with liberty starts on immortal. I'm usually stuck with too few unique luxes to keep expanding or no where left to expand. And if I get stuck at 4 cities because of happiness, I'd have rather gone Tradition.
That said, I focus all my immortal early game on catching up in science. Beelining Philosophy, then Universities, and Public Schools and being able to work the specialists. For NC, sometimes I'll settle one additional city and rush buy the library, then build the NC, then build my other (or two) cities. Hopefully there is space left.
Farm The World! You want to keep your cities happily growing. Having a size 8 capital compared to the AI's size 20 capital t100 is not what you want to see. You really need to send one or two caravans/cargo ships to the capital after you make your last settler to compensate for all the lack of growth while you were expanding.
If I can keep growing, usually having to move food routes around some, I usually start seeing the light about the time I get the public schools built, the first 2 policies in rationalism open, and 3 or 4 academies planted. I'm also able to get two or three RA friends on a standard map and I buy RAs until I'm ahead. Observatories are a lot more important on immortal, so think about that when you settle.