Courier Chess introduced what is now called the bishop in modern chess. The rook and horse/knight also have the same moveset as their modern chess counterparts. Pieces that separate Courier Chess from modern chess are the Mann, Jester, Elephant, and the Queen which retains its much more limited moves from Shatranj:
Courier Chess has an obligatory starting sequence of moves called the "joy leaps" that advances the rook pawns and the queen pawns 2 spaces forward and the queen herself one square forward. Modern players can opt not to do this starting sequence if they wish to explore other openings. However, only when performing the joy leap sequence is the Queen allowed to move directly forward two squares.
This is a summary of the pawn promotion rule as described in A World of Chess by Jean-Louis Cazeaux and Rick Knowlton: When a pawn reaches the final rank it becomes temporarily immune from capture. In order to promote to a Queen, the pawn must make 3 two square leaps backwards along its file: to the 6th, 4th, and 2nd ranks. Upon reaching the 2nd rank it is promoted and gains the Queen's moveset. The leaps do not need to be taken on consecutive turns. The pawn is not immune from capture except on the 8th rank. The pawn cannot attack while performing the leaps, that is the square it is moving to must be clear as well as the square it is jumping over else the pawn is blocked.
The above pawn promotion method is also mentioned by R.C. Bell in Board and Table Games From Many Civilizations Volume 1. Bell notes that the pawn promotion rule is actually not known for sure and the above is a known promotion sequence from another chess variant that would have been played in the same time period as Courier Chess. He does not name the other variant, just that it was known to be played in the German village of Ströbeck.
If desired, one can use a much simpler pawn promotion method such as immediate promotion to the Queen upon reaching the opposite rank.