I do know that in Emperor difficulty I try to expand as quickly as possible because I feel like the AI is expanding so quickly and also I feel like I need more cities to stay competitive in science. Is that a mistake?
Emperor is a big jump and typically if you play higher you are forced to play a "pure efficiency" game. It is the highest I can play and stay peaceful which i suspect is your game style.
Expansion is fine, it is about quite a few things in the early game and getting the order right, A few things that may help.
Layout is a lot and a bad start means a waste of a few hours of your valuable time. It is worth spending a wee while at the beginning getting a fair map.
I love scouts because of the bonuses in villages and also understanding the map is part of my strategy. However scouts are not that important to many but at least 2 slingers in the first 4 builds is a must for defense and camp clearing. getting the eureka for them is also good as are all eureka's.
Grouping your first few cities so they get shared factory bonuses is very important. production is important so do not leave this avenue too long. The price of districts does go up the more you learn (science and civics).
Workers chop down wood to rush things which is fantastic, a little OP but it helps a lot to speed up your start and you need that on emperor.
Internal trade routes with the right districts give great bonuses... i.e. built the trader at the capital but send it to the new city first to trade back to the capital to get great bonuses.
At emperor it is important to understand that the civs start with bonus units and have good growth bonuses also. This is to counteract your skill level with such things as Eurekas. The AI is rubbish at these, so try to get most things as Eureka's and that may mean half learning them then changing until you can get the eureka.
The AI will also stop at 4-5 cities for quite a while and are slow and inefficient in developing these. More than 5 cities by turn 150 would be good and have some adjacency bonuses. Commercial and industrial are the best all round flexible for all scenario districts but you do need to scatter others in there.
the AI is also more aggressive the higher you go so expect to have war declared against you unless you do everything possible to avoid it. Wars do in fact work to your favor, when they declare you should have some archers and a few other units to put in front of them. Wait a few turns to see if they attack you and if they do not, consider attacking them, they declared after all. But the secret is to kill their units, ideally defending on your soil so they suffer the worse as a nation for it (warmonger penalties) As time goes on they start to suffer more than you and start asking for peace. If you have enough amenities you can hang on until they give you a city.
Alternatively many are warmongers here and just go full out kill early take em out... not my style.
Slingers, chop, eurekas, traders, production and expand.
And that is the crux of it. You have to follow some basic rules but there is room in there for your own style. Be adaptable to your environment and situation and efficient with your play and you will win. and once you start winning you wonder why it was so hard... until you try deity. Deity is eminently beatable but requires a lot of efficiency and violence.