Start with The little virtues. Eleven essays, including one on the death of her husband in a fascist prison, written in a register that refuses to raise itself even there. If you do not like this book you will not like the novels, and you will have found that out in an afternoon rather than across six of them.
The obvious alternative is Family lexicon, which is her most famous book and the one the recent reissues push hardest. It is a better book than The little virtues and still the wrong place to begin, because Ginzburg's manner is flat, plain and apparently artless, and the whole emotional force sits in what she declines to say. The essays put that method in the open, where a first-time reader can see it working. The novels hide it.
The family book
Family lexicon reconstructs her own family entirely through the phrases they repeated — catchphrases, insults, the things her father said at dinner for thirty years. It won the Strega Prize in 1963 and it is unlike any other memoir, largely because she is barely in it. Fascism, the war and the losses of those years pass through the book without ever being announced.
Read it second, once you can hear how much the flatness is doing.
The short novels
The road to the city ; and, The dry heart is one volume containing two early novellas. The Road to the City is the earlier and slighter of the two. The Dry Heart is the reason to buy it: it opens with a woman shooting her husband between the eyes, and then explains, calmly and in order, how an entirely ordinary marriage arrived there. Together they run to a couple of hundred pages. If you want a single Ginzburg novel to test the fiction, this is it.
Voices in the Evening is a village and a family after the war, told almost entirely in overheard talk. Almost nothing is narrated; people speak, and you assemble the losses yourself. It is the purest example of her technique and the one most likely to feel like nothing is happening if you have not read the essays first.
The longer work
All Our Yesterdays runs from the late 1930s through the war across two families, and it is the most conventionally novelistic thing she wrote — the closest she came to a book with a shape a reader will recognise from other novels. Take it when you want more room than the short books give.
Happiness, as Such is told almost entirely in letters, about a son who has gone and a family that keeps writing to him anyway. Late, funny in a bleak way, and probably the best of the fiction after The Dry Heart.
Where to stop
Two books is a real education here: The little virtues and Family lexicon. Most readers who go further do so because the voice has become addictive rather than because there is a story to finish — nothing connects these books, and no volume depends on another.
If you want the fiction and only the fiction, take The road to the city ; and, The dry heart first, then Happiness, as Such, then All Our Yesterdays. Skip nothing on grounds of quality; the books are short and the level is remarkably even. More on the Natalia Ginzburg hub.
Follow the full ordered path here: Natalia Ginzburg Reading Order: Where to Start with Her Essays and Novels.