For those unfamiliar Puerto Rico has received a fair amount, but not a huge amount, of in depth analysis, That includes a fairly detailed analysis of the opening moves by the BoardGameGeek user Alexfrog. This is more than ten years old now so is fairly well known, but if you are unfamiliar his analysis suggest players should broadly prefer what he called the Main Line (and in fact they do), which runs as follows in the first round:
1: Settler (1: Quarry, 2/3, Corn/Coffee, 3/4/5: Coffee/Tobacco)
2: Builder (2: Small Market, 3: Small Market, 4/5: Construction Hut, 1: Pass)
The strength of the Main Line is that it is relatively easy to learn and remember. If a player breaks order earlier later players just take the earlier role. The most common deviations from the Main Line on this site are players who take Builder (Small Market) as first player rather than settler, and 4/5 players who take Craftsman/Captain for early VPs.
There is no similar detailed analysis for the second round, which in any case is likely to diverge based on player count. By extension of the logic that underpins Alexfrog's analysis the three player continuation of the main line should probably proceed:
2: Settler (2: Quarry)
Though I frequently see this (in my opinion) mis-played by even fairly experience players with the second player opening Trader.
I leave it open for others to comment.