Start with The Strange Case of Dr. Jekyll and Mr. Hyde. It runs to about seventy pages, and it has the odd distinction of being a book almost everyone thinks they know and very few have read — including the fact that it is built as a detective story in which the transformation is the concealed solution, not the premise.
The obvious entry is Treasure Island, and for a younger reader it remains the right one. For an adult it is the weaker first move: it is a superb adventure novel that will confirm exactly the impression of Stevenson you already have. Jekyll and Hyde will not.
The novels, in order
Treasure Island comes second, and it is worth reading as an adult precisely because Long John Silver is a far more unstable creation than the pantomime version suggests — the book never resolves whether he is a monster or the most likeable person in it.
Kidnapped is the best of the adventure novels and the one where the historical material — the aftermath of the Jacobite rising, the Appin murder — carries real weight. Its sequel is Catriona / David Balfour, and the double title is not a catalogue error: the book was published as Catriona in Britain and as David Balfour in the United States. It is one novel with two names, and readers buy it twice with some regularity.
The master of Ballantrae is the darkest and, for a lot of critics, the best: two brothers, one of them charming and malignant, pursued across decades and continents to a genuinely strange ending. If Jekyll and Hyde interested you for its moral machinery rather than its horror, go here next.
The travel books
Travels with a donkey in the Cevennes is the famous one and is much stranger and funnier than its reputation as a gentle walking book. An Inland voyage precedes it — a canoe trip through Belgium and France, and the slighter of the two. The Silverado Squatters is the California book, written while he was squatting in an abandoned mining camp, and it is where the American material starts.
Shorter fiction and verse
New Arabian Nights collects the early stories, including the Suicide Club sequence, and it is where you can watch him inventing the modern short thriller. A Child's Garden of Verses is the children's poetry and is better than the category implies — several of the poems are about illness and isolation, written by a man who had spent much of his childhood in bed.
The beach of Falesa is the late Pacific story and marks the real break in his work: a trading-post narrative about colonial exploitation with no romance in it at all. Its first publication was censored — the fraudulent marriage contract at the centre of the plot was cut — and some cheap editions still reprint the bowdlerised text.
The end
In the South Seas was assembled after his death from his Pacific letters and notes, and reads as material rather than as a finished book. Weir of Hermiston is the last novel and it stops in mid-sentence, unfinished, because he died while working on it. A number of critics believe it would have been his masterpiece. That judgement is unfalsifiable, and the fragment is good enough to make it plausible.
Where to stop
The Strange Case of Dr. Jekyll and Mr. Hyde, Kidnapped and The master of Ballantrae are the core, and three books is a fair Stevenson. Add Travels with a donkey in the Cevennes if the voice appeals, and Weir of Hermiston only if you can tolerate an ending that is not there. His critical standing was low for most of the twentieth century, when he was treated as a writer for boys; the reassessment that put him back among serious novelists is comparatively recent, and reading in this order is the fastest way to see why it happened. The full sequence is on the path below.
Follow the full ordered path here: Robert Louis Stevenson in Reading Order: Beyond Treasure Island.