If I survive I usually manage to grow my cities and economy quite well. There problem is usually money. that is cramming out so many units but the money meter in red until you get markets up.
I find the solution to this be partly the GA from liberty and to time it to last about until u get markets up. The other thing is how much u can deley the markets from going up, that is taking currency before or after LS.
Except for some civs where the UU makes it favourable for a different tech route I usually try to get workshops (and the tech for it) as fast as possible. Basicly go there directly after IW plus the necessery lux tech.
Having WS up in all cites cost money but ads a huge % of produccion making it possible to keep up with army prod but at the same ime cram out buildings. I find it good to get catas anyway and mabye a small chanse to steal HG so that one u ll want anyway. That is only one tech from markets so what I usually do, all thought it s a small gamble, is going for currency before LS, with WS up in all citie I can get markets up very fast and get the economy going t an early stage. As soon as the markets are up u ll be able to afford both army and happines buildings. About at this stage I also tend to get libraries and nat collage. If I get a window I usually produce library 1 turn from fnishing in all cities so that when markets are up I can just use one turn to get libraries in all cites and start nat collage in cap. the reason for not having them out earlier is the money issue and they give quite little tech in the satelite cities anyway when they r so small.
Another question is when to start the road network. It is imprtant for growth (happines) and transporting troops but it do cost a lot of money when the cities are size 1 and 2. What I do is usually start the roads as soon as I m finished with improving all tiles worked by cities. Starting from the capital and outards of course. Also letting the cities closest to cap grow first as soon as happines gets into + to get minimal gold loss from roads.
I dont know how u guys do. if u do the same or different....