Expansion phases, for me [Emperor/Immortal, Standard, Standard, Continents].
1. Ancient Era: Throw some elbows. This 1st expansion phase is usually the seizure of at least 1 city from each of my immediate neighbors. I like to do this before warmonger penalties accrue and before the enemy can get walls up everywhere. Sometimes, this leads to wiping out one of my neighbors. In my most successful games I take over my entire continent in this phase, wiping out my rivals before they build a 3rd city (that's only happened a couple of times - a more typical result is to knock my neighbors down and become the major power before warmonger penalties arrive and I have to be more circumspect).
- At Emperor and above, the AI starts the game with a Settler. So a 2nd city is not evidence that an AI is expanding. Don't freak out when you see a non-capital city. Just take it.
2. Ancient to Classical Era: Lay down borders. The 2nd expansion phase is building settlers. I adopt Colonization and build multiple Settlers simultaneously. If everything has gone well thus far, this is 3-4 new cities. I put these around my periphery, claiming as much land as I think I can, trying to box out my neighbors.
- Make sure you've researched Bronze Working, so you can see where the Iron is.
- This is a good time to claim Natural Wonders and look for good locations for coastal cities.
- You'll probably be pushing against a neighbor here. That's part of the point. Found multiple cities on the same turn, so that when your neighbor gets upset you can tell them "Okay, I won't settle near you any more."
3. Push borders & fill in the gaps. The 3rd expansion is similar to the 2nd. If I can expand my borders, I do. If I'm already crowded, I look for gaps and places to add a city within my existing borders. Sometimes there's another war of conquest here, but by this point I'm usually looking at warmonger penalties, so I have to think about it.
- Switch to Settler View to see where grey and green hexes are, between the red hexes of your cities. Fill in those gaps, even if the new city is in a poor location and won't be very good.
4. Start an overseas colony. This expansion phase comes after I discover an unsettled land mass or an AI that's turtled. Colonization again and another round of multiple, simultaneous Settlers. This time I have some Builders and a colonial army ready to go with the Settlers, and I save up several thousand gold in preparation (to claim hexes and purchase buildings). I like to get at least 3 cities founded together, so the colony can be a little bit self-sufficient. Usually this doesn't involve waging a war, since the whole point is to seize territory the AI has overlooked or hasn't discovered yet.
- Don't forget to build a navy, unless you're the Suzerain of Lisbon. You'll want to use domestic trade routes to boost the food and production of these new cities.