Fin, welcome to the forums.
My view after playing the game many years it to find Snaaty's excellent post and read how he starts a game early (just his early part). It's great stuff and has greatly improved my game.
What to start first depends on starting techs, map, resources. Here are some hints
1) You should grow your city to the happiness cap 90% of the time.
2) Your first build is dictated by the available resoruce. If you have fishing and available seafood build a workboat ASAP, even if you only work a 3 hammer forrested hill. If you start with agriculture, build a worked unless your in the middle of a forrest. 80% of the time I start with a worker, and those games I usually start off very strongely.
3) Your first tech should be a food tech unless you start with several like the Americans. If you have agriculture or hunting and there are cows or pigs in the BFC then Animal Huisbandry is very powerful as it opens up horses.
4) Once you are satisfied with the food situation, shoot for hunting and archery and mass build archers until you hit the happy cap. Others may say mining/bronze working and sometimes this is OK, but 90% of the time go archery, trust me on this.
5) Once you are at the happy cap, build your settler and more workers/settler from the capital while your other cities build monuments and barracks. Sometimes you need to get either the wheel or sailing to communicate the cities.
Some leaders need to be taken with a grain of salt and do not fall into the above ritual. One I had alot of problems with was Charlemange who starts with hunting and mysticism. Temtping for an early religion but you have to be willing to expand slowly, not always a bad think as I finally found out to win.