Start with Dragonflight, the first one published. This path is publication order and that is a deliberate choice, not laziness, because Pern is the series where the publication-versus-chronological question has an actual right answer.
Here is why. Pern looks like a fantasy world — dragons, holds, feudal ranks, a planet-scale threat called Thread — and it is not. It is a lost human colony that has forgotten it is science fiction, and working that out alongside the characters is the entire pleasure of the first three books. Chronological orderings open with Dragonsdawn, which tells you on page one what Dragonflight, Dragonquest and The White Dragon spend three novels carefully revealing. There is no way to unspoil it and no compensating benefit.
Dragonflight is around 300 pages and moves fast. Give it a week.
The original trilogy
Dragonflight, Dragonquest, The White Dragon.
Dragons are the planet's only defence against Thread, bonded to their riders at hatching. Dragonquest is the political sequel and a much better problem than another monster: five hundred riders pulled forward in time who do not fit the present. The White Dragon is where the clues McCaffrey has been planting since chapter one — the abandoned rooms, the records nobody can read — finally cohere, and the science-fiction premise becomes explicit.
One honest warning. Dragonflight was published in 1968 and its sexual politics show it: the mating-flight scenes and the treatment of Lessa involve coercion the text does not treat as a problem. Readers regularly name this the hardest thing about the early books, and it does not recur at the same pitch later.
The Harper Hall trilogy
Dragonsong, Dragonsinger, Dragondrums. Written for younger readers, short at 200 to 260 pages each, and the ones most readers press on friends first.
Dragonsong and Dragonsinger are essentially one book split — the first ends on Menolly's arrival at the Harper Hall and the second begins there — so read them back to back. Menolly is the series' best character and its only real portrait of talent being actively suppressed: a girl forbidden to make music in a hold that needs a harper. These books also show Pern from ground level, with sea holds, fishing, hunger and rank, which is the social texture the dragonrider books skip. Dragondrums overlaps the events of The White Dragon, which is why it sits after it.
Back into the past
Moreta, Nerilka's Story, Dragonsdawn.
A buying note: Moreta is catalogued in many records under that bare title though it is normally subtitled Dragonlady of Pern. Same book. Nerilka's Story is very short at around 180 pages and should be read immediately after as its companion — the same epidemic from a hold woman's viewpoint, unglamorous and practical, a deliberate corrective to the heroic version. Moreta itself works because you have already heard its events sung as a ballad in the earlier books, so you know the ending before you start.
This is where publication order pays off most visibly. Dragonsdawn sits ninth, not first. It is the colony landing — the ships, the gene-splicing, the deliberate creation of dragons — and read as an answer it is thrilling. Read as an opening chapter it would have destroyed the first three books.
Closing the arc
The Renegades of Pern, All the Weyrs of Pern, The Masterharper of Pern.
Read the first two close together; the first sets up the second directly. The Renegades of Pern follows the holdless — outcasts, traders and criminals — which is the only sustained view of Pern from outside its institutions. Then All the Weyrs of Pern inverts the entire premise: the colonists' artificial intelligence is reactivated, technology returns, and Thread becomes a solvable problem. Ending Thread permanently is what the whole sequence was built toward, and the dragons' purpose is completed rather than continued.
The Masterharper of Pern is a late prequel covering Robinton's life up to the opening of Dragonflight, which closes the loop on the character introduced in book one. Reading it twelfth rather than chronologically is the same principle as Dragonsdawn: it is worth far more after eleven books of loving him.
Where to stop
All the Weyrs of Pern is the intended conclusion and stopping there is entirely reasonable. The Masterharper of Pern works as a coda. Beyond these twelve the Pern shelf continues well past what McCaffrey was building toward, including volumes by Todd McCaffrey; this path stops where the story actually finishes. For more series with the stopping point marked, see the rest of the site.
Follow the full ordered path here: The Dragonriders of Pern Reading Order: Where to Start.