The early game is the most crucial point so it would be very helpful for you to post some more info, perhaps a save file, about your typical early game.
Some things that helped me advance:
1. Get a better understanding of city placement strategy. Your early cities should have a good balance of food and production. Think of each food or hammer as 1 yield. The best cities will have multiple tiles in their immediate borders with 4+ yields (ex. 2 food, 2 hammers or 1 food, 3 hammers). Of those you'll want at least one that is food focused (ex 3+food or a resource that can be a good food source such as wheat, rice or sugar). When placing cities also be cognizant of what resources can be quickly improved in that city, bonus resources for needed tech boosts, strategic resources for military, luxuries for amenities.
2. In the early game it's important to "forward settle" the AI; settle your cities as close as possible to them without incurring loyalty penalties and settle in a way that blocks off the AIs potential for expansion. You can then fill in the gaps with cities during later expansion. Use governors strategically to address loyalty issues.
3. Have your empire go through different building phases. For example, hold off on building lots of settlers until you have the "+50% production to settlers" policy card, the government plaza with ancestral hall and the governor Magnus with the Provision promotion (Settlers do not consume a population). Once you have those in place, build as many settlers as you can. Do the same thing with military; build only enough units to defend until you have the appropriate cards and an encampment with the appropriate building, then rush build a ton of them. Same thing with builders; when you get to feudalism, put in the policy card to get more builder charges, place Liang in a city with high production, then pump out as many builders as possible.
4. Diplomacy with the AI is crucial, especially early. Make sure to send a delegation to every AI you meet ON THE TURN YOU MEET THEM. If you wait even one turn, the AI could move to unfriendly status and won't accept your delegation; the AI will always accept your delegation on the turn you meet them. The delegation is important because it improves your relationship with that AI, giving you better trade deals and making it more likely for you to declare friendship. You want to declare friendship with every AI you can unless you plan on going to war with them within 30 turns (friendship lasts 30 turns and you can't declare war during that time). Declaring friendship will do a few things: -it will eliminate the possibility of that AI declaring war on you (see above), -it will increase your relationship with them, getting you better trade deals, -it will open up the possibility that for alliances, which can give you big boost to science, culture and gold.
5. Trade with the AI regularly, particularly your open borders. The AI will almost always pay you for open borders, so make sure you try to get gold when offering open borders to them (which you should do ASAP as this also increases your relationship with the AI). The trade screen process is currently a tedious nightmare, so unfortunately you have to do a lot trial and error clicking but in the long run it's worth it. If every AI only gives you 1 gold per turn (GPT), then you're getting +7 GPT very early in the game, which is a HUGE boost. The AI is likely to give you even more GPT though, so you can realistically get 10-15 GPT, plus some one time gold off them, giving a huge boost to your early economy. Regularly shop around your extra luxury and strategic resources as well. These extra sources of gold are essential as your empire expands.
6. Districts and placement. Place your districts early, meaning that you pick the spot they should go and the city starts building them there, but don't finish building them until later. District costs increase as your go through the tech/civic tree but the cost is locked in once you place them, even if you don't finish building them. You should hold off on completing too many early districts, really only focusing on the government plaza with ancestral hall, a holy site for an early religion/faith for golden ages, and then a few choice ones, usually campuses. You should try to place your districts as early as possible, so keep an eye on your population in each city. Your early priorities after the previously mentioned ones should be campuses, commercial hubs/harbors (for trade routes), then whatever districts you need to build for tech/civic boosts.
7. Settling. I usually try to settle 2-3 cities very early. My starting build order is usually warrior, settler, slinger, settler, scout. My second and third cities will build some combination of granary, monument, trader, districts and military depending on the circumstances, with one of them focusing almost exclusively on military if I'm going domination. My capital will usually then focus on getting government plaza and ancestral hall, then pumping out settlers with the goal of getting to 10 cities by turn 100, either through peaceful expansion or conquest. I'm almost always able to get to 8 or 9 by turn 100 and if the cards are right I can get to 10.
Hope this helps. I meant for this to only be a couple of quick tips but it gradually got longer and longer as I typed.
