I was a long-time Civ3 player, but didn't play Civ4 or Civ5, and only got Civ6 in 2019 after some life-changing experiences. I don't play Civ6 any higher than Warlord, mostly because I want a low-key easily enjoyed game.
When I started playing Civ6, I remembered one key thing from III - Population = Power. So I build cities faster than the AI (usually).
In my games, I start by sending my initial Warrior out exploring my immediate environs, settle my first city, and start building a Scout. The Scout goes out to explore, and I start on a Slinger. Once the Slinger is done, I put him on alert, and start a Settler. ( I may need to build another unit before the Settler, depending on how fast my city grows.) I also research Archery as soon as practical, to get some stand-off power. Once the Settler is built, I send him out, accompanied by the Slinger, to settle a new city. From that point, my build order is Slinger/Archer - Settler, until I have six or more cities (depending on the map). (Settlers are always accompanied by a military unit!) I don't explicitly make Builders at this stage unless there is some overriding reason. Besides, I often get one from popping a goody hut.
I use my Warrior, sometimes with the assistance of a Slinger, to deal with Barbarians. Particularly with Barb Scouts, if they poke their noses into my cities, the guarding Slinger will knock them down, and the Warrior will finish them off. Soon, the Warrior gets his first promotion, which I make Battle-Cry (+7 against land units). With that, he can stand up to the Barb Spearman in the Camps. (If the Barbs get Horse Archers too soon, though, I often will turn tail and hide, until I can get Archers and Swordsmen. Discretion is the better part of valor.)
While my capitol is busy churning out Settlers and their escorts, my other cities usually start with a Monument and a Granary (not necessarily in that order).Then, they start building districts and their associated buildings, and Traders. Yes, I often don't have many (any?) districts built before the Classical Era. By this point, if it looks like it's going to take "forever" to build a district in a city, I will prioritize Builders, to build Mines, Quarries, Lumber Mills, etc., to increase my production.
One thing is, you have to build and maintain military units. I generally have a bit more than one unit per city. This allows me to handle the Barbs as needed, and also puts me at, or at least near, the top of the pack for military strength. This keeps the AI from doing a surprise DOW on you. I also tend to prioritize Gold on my trade routes, which helps me afford to keep my military strong.