Start with The Posthumous Memoirs of Bras Cubas. The narrator announces on the first page that he is dead and is therefore free to be honest, and the novel proceeds in 160 very short chapters of digression, self-justification and cheerful malice. It was published in 1881 in Rio de Janeiro and reads like something written a century later. Nothing else in Machado is a better introduction, because nothing else makes his central quality — an irony that never resolves into a moral — so immediately obvious.
The alternative usual first choice is Dom Casmurro, his most-taught novel and the one with the famous unanswerable question at its centre. Read it second. It is the better-constructed book, but its whole effect depends on the reader distrusting the narrator, and that distrust is a skill The Posthumous Memoirs of Bras Cubas teaches you in a much more entertaining way.
The mature novels, in the right order
Three books form the core, and they were written in this sequence: The Posthumous Memoirs of Bras Cubas, then Quincas Borba, then Dom Casmurro. They share a philosophical joke — a doctrine called Humanitism, invented by a character who may be insane — and reading them in publication order lets the joke develop.
That said, most readers do better with The Posthumous Memoirs of Bras Cubas followed directly by Dom Casmurro, saving Quincas Borba for third. Quincas Borba is the weakest of the three and the most conventional in shape, and putting it between the two masterpieces slows people down at exactly the wrong moment.
Dom Casmurro is where Machado turns the technique on a single relationship. An elderly man narrates his marriage and his conviction that his wife betrayed him with his best friend. The novel supplies no evidence either way, only his account. Brazilian criticism has argued about it for over a century, and the argument is the point.
The late novels
Esau and Jacob and Counselor Ayres' Memorial are quieter, drier and set against the end of the Brazilian empire. They share a narrator and are best read in that order. Neither has the energy of the mature trio, and both are excellent — but they are books for a reader who already knows the voice and wants to hear it grow old.
The shorter work
The Alienist is a novella about a doctor who builds an asylum in a small town and gradually concludes that everyone in it belongs inside. It is around eighty pages, it is very funny, and it is a legitimate alternative starting point for anyone unwilling to commit to a novel first.
The collected stories of Machado de Assis is the significant recent event in his English reception: seventy-six stories in one volume, translated by Margaret Jull Costa and Robin Patterson. He was a great short-story writer as well as a great novelist, and several of the best pieces here rank with the novels. Read it in fragments alongside everything else rather than straight through.
What to skip, and be honest about it
Resurrection, Hand and the Glove and Helena are early novels from the 1870s, written before the change in method that made him important. They are competent romantic fiction. The catalogue lists Hand and the Glove under a scholar's name rather than the author's, which is a filing artefact rather than a different book.
Read them only if you become interested in how a conventional novelist turned into an experimental one at the age of forty. As a starting point they are actively misleading, and a reader who begins with Resurrection will not understand why anyone cares.
A translation warning worth heeding
The Posthumous Memoirs of Bras Cubas has appeared in English under more than one title — an early and long-standard version was called Epitaph of a Small Winner, which is the same novel. There are several modern translations, including a heavily annotated Penguin edition by Flora Thomson-DeVeaux; the notes are genuinely useful, since much of the comedy is aimed at nineteenth-century Rio. Check the title page before you buy so you do not end up with the same book twice. The full reading path sets out the stages with study notes.