Man, this one messed me up for ages because the series uses two different titles for the same first book! The absolute starting point is 'Murder on Astor Place'. That's the novel where we first meet Molly, a young Irish immigrant in 1901 New York who gets tangled in a high-society murder after she witnesses a confrontation.
I started with the audiobook, thinking it was a later title. The confusion comes because some editions or databases list it as 'Death of Riley', but that's actually book two. 'Murder on Astor Place' sets the whole thing in motion—her fleeing Ireland, meeting the detective Daniel Sullivan, and stumbling into her first case. It's the only logical entry point to understand her character's journey from a scared refugee to a determined investigator. After this one, the chronological order is pretty clear in the titles and numbering.
I'd double-check the cover of whatever copy you pick up just to be safe, but start there on Astor Place.