The automated city management seems to value total yield on a tile most, so your first citizen may be set to work on a 1F 1P 1Science tile rather than a 2F 1P tile which will halve the time it takes your city to grow to double size. Use the city management screen to look at the tiles being worked and alter them (temporarily) if necessary. Maximise your production by getting a builder. A farm on grassland rice will get you 4F and turbo charge city growth. Place your cities so that they will grow (eg plains/tundra cities are hard to grow) and if necessary buy a high yield tile when you have 50G.
You can buy your first settler for 320 gold. If you have gold yielding resources (like copper), sell a couple of resources to other civs, pop a few goody huts for gold and clear a couple of barbarian camps, that total can be reached quickly. I'll build a settler rather than buy one though, if I'm not rich.
Some might consider this an exploit, but if your scout or warrior hangs around a neighbour's city, you can often capture a settler. You see the population drop in a city and you know it's produced a settler. Look for it, as they often get sent out alone, or separate from the military escort at some point. You do need some practice in this because sometimes 4 warriors will descend on you and your captured quarry, but most of the time its easy to get away.
I usually buy or build one settler before researching Early Empire, but once I have that, my settler building only happens when my Government runs the Colonization policy. You will then build settlers faster or you can chop down woods, harvest stone etc and get a 50% boost to the yield. If playing R&F+, appoint Governor Marcus and you get a 50% bonus added to chops. Marcus also has a promotion that means your city population doesn't decrease when you build a settler. In R&F+, if you get a golden age, then you can choose monumentality which means you can buy settlers more cheaply and with faith as well as gold. Again in R&F+, you can build the Government Plaza and then ancestral hall.
Since your OP,
Knighterrant81 has produced this guide:
https://forums.civfanatics.com/thre...he-decidedly-average-feedback-welcome.644711/
You might also look at some you tube vids. In his one, (which is deity) civtrader6 holds back Norway with 2 warriors and builds lots of cities