Chopping 2 Settlers is jumping the gun a bit, first off.
But your real problem is the economy. My advice is to spend more time in the city screen looking at just how much commerce each city is producing. More importantly, look at where that income is coming from.
A city founded near a gold mine, or a commercial resource on a river, will immediately pay for itself. You don't want to found cities that can't pay for themselves until you're running a surplus in your other cities.
Are you building cottages early? Are you building a few and then forgetting to keep building more? You can never stop increasing your cottages (or specialists) or you will fall behind. The details of how to do that are fairly straightforward, if you pay attention to the food and gold in each city.
You should definitely be doing things like, when a city hits the happy cap put all the workers on gold-generating tiles. This may mean 2 cottages and the rest Coast, so that your production virtually stops. That's ok. Get your key buildings up early and then focus on squeezing as much gold as you can out of your commerce cities - within the happy cap.
(This is why Slavery is so powerful. Say your happy cap is 6. Use the whip to quickly raise a Granary, Library, and Courthouse. Then stop working those mines and make everyone work commerce tiles. This city is now generating a healthy surplus. The next step is to increase the happy cap. And when it's time for more buildings, you can always go back to working the mines and whipping people. After you cap out again, go back to gold tiles. You get the picture. Once you have a metropolis you convert all the mines to windmills and rake in the gold.)
This is why raising your happy cap early is also important. It's easy to get a good production city going early. It's harder for commerce (usually) because cities need to be big to generate a lot of gold. So if your population is stagnating, that means your economy is too.
I also think less-experienced players tend to overlook opportunities to generate gold in the very early game - before cottages or specialists even come on the scene. A gold mine is an obvious boost, but consider Ivory on plains next to a river. This tile generates 1F1H2G without requiring any improvement. If the city also has a food resource, you can generate 4+ gold as soon as it hits size 2. This is the type of thing I might do in the short-term to keep my research going. Eventually I'll want a better source of income, but that kind of micromanagement can keep your economy afloat while you expand. Lots of little changes add up.
Finally, there are trade routes. In certain situations, wonders which add trade routes will be the best thing you can do for your economy. For example, the Great Lighthouse if very easy to beat the AI to, and it's great for the economy if you're on an island. But for the most part these play a small role in the economy until the mid to late game.
So it's actually pretty simple: keep adding cottages or specialists, keep raising the happy cap, and your economy will grow. Doing this faster and better is what comes with experience. I think for most people, at some point it "clicks" and they realize just how to manage their gold. Civ can be difficult because information is scattered all over; but for running a good economy I think the city screen is most important.
Specialist Economies are a different beast, one that I wouldn't try to tame until I had mastered the cottage economy. (But still run a GP farm, of course.)