I agree, it's good to let your city grow a bit before building workers.
One slightly more aggressive method, although it might take a bit of getting used to.
I normally don't build a worker, and sometimes don't even build a settler. As a worker cost 3 warriors, a settler costs over 6 warriors.
Starting a build queue with 3 warrior’s means you’ll have 3 warriors patrolling 3 Different AI boarders. You have them there long before the cities expand to the 3rd ring, so you’ll get some opportunities for a worker steal, leave your warrior just outside their territory just next to a resource tile in their second ring, when they go to work that tile you can declare and take their worker(note in the early stages they leave workers unguarded, later on you'll need 2 warriors as they sometimes escort the workers). There, you’ve just paid for your 3 warriors, and you get a worker about the same time you would if you built it. Meanwhile your city has been growing, so getting more productive. When guarding the boarders of 3 AI, you’ll often get the opportunity of snatching 2 workers, which is twice as effective then building them. On top of this the early loss of the AI worker really cripples the AI, making later warfare much easier. Don’t wory the AI never really retaliates in these early wars, the most they will do is send a warior or two, but in the time that you would have built your first warrior and settler you built 9 warriors, so retaliation is just an opportunity to feed more experience to your units. And remember, every unit they send out of their city is one less unit you’ll have to fight against the city culture.
Building 6 warriors is less work than building a settler, the AI normally only has 2 warriors, and if not creative then will have 40% culture in the capital and 0% culture in extra cities. In most of their cities they don’t have culture, so fighting 2 of their warriors with your warriors is about 50-50 chance (actually better after c1+shock from fighting barbs). So taking a city will be 50 hammers, taking their capital might take a bit more, but remember the cost of warriors is nothing compared to the cost of a settler. You can comfortable sacrifice 2-4 warriors to soften them, and then finish off with 2 promoted units. Hammer cost 50-100, and you’ve got their capital. Capitals also often have a lot of resources. More resources means more happiness everywhere.
All this time you can just be building more warriors to further the attack onto AI2 and AI3.
The advantage really comes after a few battles when you start getting more experienced units, then you’ll have something like C3 shock2 warriors before you know it, and they can take down any city without casualties.
This is getting off topic to early war mongering (I’ll write a spoiler to show you an example).
Anyway you should try a worker steal sometime, it gives a really powerful early boost. Just patrol the boarders of 2 or more AI as you might not always get an opportunity with one.
On the topic of later economic growth, cottages is a must, cover all the land with cottages.
going for education and writing early is good, giving you cottages, EC, libraries, and a great sage. if you're worried about happiness i guess you could use the sage for mathematics, gambling houses are very powerful, at 60% research, you have +4 happy faces, meaning all cities can be 4 larger, more production and science. although i often just get an academy, so research on better military tecs goes faster, then get the happiness from capturing enemy capitals with resources.