I walk the warrior NE. Nothing of interest, so we settle in place. Build worker first, research agriculture -> AH (deselect tech for first 5 turns in case we meet someone else with agriculture for a lower tech cost).
The worker builds a mine in the meantime, before turning his attention to the pigs and sheep after AH is in. The second build is a
settler. This is a bit of an unorthodox play I'll admit, but on emperor barbs are no problem at all so more fogbusters are not needed, and I really wanted to be able to settle the gold ASAP. Not only to beat Saladin to our only source of iron (I knew that beforehand from peeking at your screenshot - so not a blind playthrough entirely, I'll admit), but to claim the gold, which will boost our research from 11/turn to 18/turn! The settler also only takes 15 turns or so because our imp trait + PH mine gives us in reality 7 hammers/turn when constructing it. And the worker finishes the pig just about in time for the settler to finish, allowing us to grow fast, building a warrior next.
Research-wise, after AH, I go to bronze working. BW is important because it allows us to chop and whip - the two major sources of production early and mid game. Then I go pottery for cottages and granaries, writing for OB and libraries, hunting to hook up ivory, and straight to IW for our UU. After IW we get math for better chops, so that every forest is worth 50% more.
I walk the first worker to the site of the second city to immediately start mining the gold. After building the warrior, at size 2, the capital immediately starts with another worker. There are a lot of tiles needed to be improved, and as such the worker is needed.
I switch into slavery after the worker is out, and start on a barracks until size 3, when I
switch production to another settler. That settler is partly chopped, and goes NW to claim the cow, flood plains, and copper. Meanwhile, the first worker, having finished the gold and rice, is roading back to the capital. After that settler, progress on the barracks continues before another settler is whipped for 2 pop - claiming the ivory, corn, and gold further north. This is the last settler we will build for quite a while. The overflow goes into a granary, which is then completely with more overflow from worker whips. Meanwhile the other cities build granaries and barracks.
As soon as IW comes I start whipping and chopping praetorians like mad. This is where the barracks and granaries we built come in handy - each praetorian I build starts with one promotion, and every whip takes half as long to regrow from. The workers, besides chopping, help with roading to Saladin, our target.
By t90 I have about a dozen praets, and declare. By my calculations I have built about 12 units in 15 turns with mostly whipping popuation and chopping forests - something that is not only possible but a skill necessary on difficulties above emperor where just raw hammers will never outproduce an AI (again, this require a bit of finesse so if you want turn-by-turn analyses of when I chopped and whipped, feel free to inquire further).
Saladin is caught completely off guard, and his first 2 cities fall easily. Praets are so strong because their high base strength makes them almost as powerful as a
maceman for only 65% of the price - meaning they can completely steamroll through anything, including axes, with city raider promotions. So we do just that, gobbling up his capital next, and then 2 other cities.
Then, we take peace for several juicy techs.
As you can see, early war pays far greater dividends than just sitting back and building random buildings. With under 10 turns of conflict, I have over doubled the land I have to work with, completely crippled a rival AI (he isn't coming back from that with just 1 crappy jungle city left),
and extorted 4 techs for free. In return I lost: a handful of units, and a potential trading partner (who won't be teching anyhow after this, and whose resources all belong to me, so...did I really lose anything?). It is now not even t100, but I have about 10 cities and am probably top in both score and military. And there are plenty of people left to kill

.
Now, I know what you are thinking. "But...the economy!" I know you brought up the issue that you seem to run out of money to expand, and I imagine you are mortified knowing that I now barely break even at 0% science. But fear not - that is all part of the plan! You see, now that I have alphabet, I can
build research, which enables me to tech even without putting any commerce into science. The gold I accumulated from setting my slider to 0% for the entire war as well as looting cities will get me through currency - after which I can build wealth to gradually turn my slider back up, and also get an additional trade route per city which is not insignificant now that I have about 10 cities. And even though new cities past #4-5 cost more than they give in return
initially, it is an investment that pays off as the cities grow, very soon contributing far more than they take! At the same time, I have started placing down cottages which not only give commerce but also will grow in time to give even more!
This last part, I think, addresses one of the core problems you are having with this game. You are afraid of expansion because you are afraid of costs and don't know how to handle your economy. But keep these tips handy and you will more than be able to hold your own in terms of research.
1. Cottages will be the
backbone of how you research for most of the game, generally speaking. Place them down on every non-resource, riverside tile, work them, and you will find your beaker rate far higher than if you just farmspam everything.
2. Cottages are also how you keep up with increasing costs. In case you haven't caught on - you aren't building enough of them.
3. DON'T build useless buildings! If you're not building the essentials (granary, forge, library, barracks if unit-building) OR military to attack someone, build research or wealth!
That is how you keep a large part of your economy going during and after conquest.
4. Don't be afraid of turning your slider back or even having to stall research for a bit to expand, whether peacefully or by the sword. The new cities will pay off very, very quickly. To show you just how quickly, tune in next time and you shall see.
Think of it as an investment - if you had $100,000, you would be understandably hesitant about parting with half of that, just as you may be hesitant to create 4 new cities which requires you to halve your research rate for the time being. However, if that 50% is into Microsoft stock in 1975 that will soon exponentially compound in value many times over, then you'd be more than happy to throw your money into it. Just as, right now, I am more than happy to eat the extra maintenance for both cities and units to have a far stronger economy than I ever would've been able to have without the war, in a very short while.