One trick worth to follow with most civs for rapid early expansion:
In ancient era (2 pop starting city size):
1. spend 5 turns with 2 workers on grass (or fish if you have it and have bronze working) to grow to 3 pop
2.a. put 2 workers on trees and 1 on grass and choose to build settler, in 5 turns you get a settler and your city is back to 2 pop instantly, so you can start all over, i.e. you get a settler out in each 10 turns.
2.b. if you have enough gold to rush settlers, then you can spend 2 turns on 2 grass + 1 tree, getting 8 food + 4 hammers, then rush a settler for 32gold, set the 1 worker on grass and next turn you are back to 2 pop, so this way the whole loop is only 8 turns.
There are some variations that make it more efficient for some civs, and also some special resource can help too, e.g. cattle if you have Code of Laws (Romans), whale if you have navigation (Spain), ox with Horse Back riding, so try to use those resources.
Romans in republic (or other civs if you get the pyramid from the Angkor Wat and switch to republic) can do it without 3pop: just collect 8 food in 2 turns on 2 grass, then spend 5 turns on 2 trees to build settler or rush earlier if you have gold.
Once you reach medieval, or if you play Chinese, your new cities start at 3pop, so you can skip the initial 5 turns of growth, so it becomes much faster to get a chain of settlers going: build/rush a settler in 5 or less turns, then the new city does the same and keep going, while the old 2 pop one is put to trade or unit production.
Americans should reach medieval and republic ASAP, then just put all workers on trade to make gold and rush a settler for 20g from each new city and move along building the next city rushing the next settler etc. This is the fastest, no time wasted on food or trees, you can easily get 20-30 cities by 0AD even in multi-player.