Not Carthage-specific, but I've been going Pottery > Writing > Writing II often when I planned to stay with 1 city for long.
I find the exact sequence is much more in-game situation dependent than in Civ VI and other previous Civ games, which I regard as a Very Good Thing.
So, while Pottery - Writing is a good general initial sequence, I frequently start with Husbandry - Sailing to both get an early ranged unit for defense against hostile IPs and to allow Scouts and other units to use sea travel which boosts my initial scouting considerably - especially on Fractal or Archipelago maps with lots of coastline.
When I already have a Science Boost, either from the Civ or the Leader or the combination of the two, I will frequently 'put off' Writing in favor of Husbandry - Masonry or Irrigation to both get ranged units early and start building up my City early - but building the city frequently depends on numbers and closeness of hostile IPs.
IF your first city is on the coast you almost have to start with Husbandry, because you will meet hostile IP Galleys fairly early and only ranged units can adequately defend against them - a pair or trio of IP galleys can and will destroy an early coastal Settlement in very short order, although I have not actually seen them do this to a Capital, only subsequent Settlements - but that I have seen frequently.
The one time I played Carthage was on a Fractal map, so I started with Husbandry - Writing - Writing II - Sailing and was pretty happy with it - had no trouble seeing off the early hostile IPs, managed to keep up with other early Science AI Civs, and had most of the initial fractal continent scouted before anybody else. On the other hand, that start was right at the epicenter of 3 hostile IP settlements, so it was a pretty combative start!
