Fast expansion is one of the key factors to faster victory times and winning Deity, and it's notoriously difficult to get right. I fail in most of my games, or at least I look back and easily see where I made mistakes. There are many approaches to settling fast.
1. Ancestral Hall: When you have tons of free land available and are expecting to do well in culture (maybe you are Rome or Greece) to get a fast Feudalism, this might be the strategy to go for.
You want to get two Settlers out very early, either through building both or via the Settler Pantheon. Your first 2 expansions should have a decent amount of production, so hills are good, as are pastures. Strategics should also be a priority. You will probably want to start building settlers as your 2nd, 3rd or 4th build (requires some decent growth, you might want to purchase a tile). Having your cities down by around t30 is key. Get a builder first in both those cities and improve their production. Meanwhile, get the capital to at least 4 pop (one for your first district, four for the government plaza). Prioritize getting the Govt. Plaza, then Early Empire, then Political Philosophy, then Feudalism. As soon as you get Early Empire, when your Workers are finished, you start producing Settlers in your expo while still working on the Plaza/ Hall in your capital. It's fine, they'll have to walk around for a few turns anyway before plopping down. After AH is build you want to spam settlers for the next 20-50 turns in your capital depending on how much space there is. Make sure you have Magnus with Provision promoted, sitting in the capital, before that first Settler from the cap is built. Make sure you have your builder charge card in before you settle your later cities (you'll want Feudalism asap, t55-75 would be ideal, but not always realistic). It can be worth to wait one or even two turns with settling a city just for those 2 charges. Monumentality is almost a must.
2. Early Empire: When you have decent amounts of free land and aren't looking very strong in terms of culture.
Ancestral Hall is not required or always ideal depending on how much space you have. With this approach you will build only one city before EE, and you'll likely build it later in order to fit a monument, which will in turn speed up EE. This way you can build your second settler with the +50% production card slotted in. Needless to say for this approach go EE over the Government Plaza. You are not reliant on getting your cap to pop 4 and you're also much more flexible in the ways you settle. Choose your city spots freely, focus on cutting enemies off and securing territory always, instead of settling cities that are convenient. Look for mountain ranges, water, city states and the like to box your opponents in and stop their settlers before they're even built. I often times run into the situation where I am seeing an enemy settler running out of their borders and mine is already in place to forward settle, that is probably the ideal scenario. With this strategy you can still go AH, or you could go Warlord's and take out one neighbor. This mixed strategy has also worked pretty well for me and is recommend for people who have already played lots of AH games and want something new. Monumentality is welcome, but not a must.
3. Domination: When you have little space, but many close-by neighbors
You usually only want to settle two cities, preferably very close to a neighbor with one good Encampment spot. After you build your cities you will likely only expand through conquest, unless you need a specific spot for a strategic. Note that this can work for science, culture, etc. games as well, not just for pure dominations. Monumentality isn't needed, but isn't bad either.
Of course there are many other ways to expand rapidly, but these are some of the more common ones and cover most scenarios.
Edit: Also when expanding builder first for new cities or monument first?
Builders when you can afford Ilkum (maybe as Greece, or when you get a lightning fast govt), or when you have very strong tiles to improve (mining luxes, horses, iron for ex.). Monument first when you are lacking culture and don't have very strong tiles to improve (farming ressources, production-less luxuries, fish or crabs).
