I started playing on emporer and I've been having some trouble adapting to the early game. In my first few games I only was able to build like 2 cities before the AI completely boxed me in. I tried again and I was able to build a bunch of cities, 6 I think, but my economy colaposed and I couldn't reshearch anything. Then I tried to hit like 4 cities early but even this ruined my economy. So what's a good number of cities to have in the early game?

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Having 2/3 city's on smaller to normal map size isn't that uncommon. Going bankrupt with 6 city's early on isn't either. Most of the time you need to expand at the cost of your neighbours. Growing steadily having the gold from captured/razed city's as a buffer for your expsenses, still maintaining a good research level. Note: you don't need to run at 100%. You are doing good if you research faster then your opponents, even if you run at 10%. (Kinda extreme example.)

My best Emperor results were from vanilla HC and WL Augustus.

Early aggression is the key with both of those.

There is a strategy guide for a more peaceful approach with other leaders somewhere in the strategy articles section.

You might want to review the Emperor's master challenge games posted be aelf.

Well the higher the difficulty level the harsher the more difficult it is to sustain large numbers of cities, especially early in the game. So you will often be better off with just two or three relatively large and well-developed cities than you would be with six cities.

Of course 2 or 3 cities will not be enough to win the game with over the long haul (on most map sizes). If the AI boxes you in then you'll just have to plan on attacking someone and gaining a few cities that way. Also, consider building cities with a little more overlap than you might normally like, and you may find there's room for more cities in your territory than you thought.

Even on Emperor, though, you'll sometimes find yourself relatively isolated and be able to expand at a measured pace. You can also try tthe usual trick of blocking off areas for later settlement with your culture and refusing to grant Open Borders.

Vanilla HC is an excellent starter for Emperor, to get used of the pace and the early war, not to forget, the earlier barbs. Financial helps you not to fall behind too much. There are various articles about a Quechua rush. I will always recommend starting with the Inca first and then move on.

There isn't a hard and fast rule about it. It depends on what kind of terrain and resources you have. If you have one or two gold mines or rivers or floodplains, that would affect how many cities you can afford to have early on.

Early aggression almost always pays off, though. You don't have to keep enemy cities. You can raze most of them and keep the better ones that are not too far away. You get plunder money, breathing space and weaken/destroy a neighbour (whom you can finish off later for more gains). Generally a good option.