There are 4 modern subseries (usually 3-4 games per each arc), each subseries shares the same setting and characters but there are no connections to other subseries. These are:
Arland series: Rorona, Totori, Meruru, Lulua
Dusk series: Ayesha, Escha & Logy, Shallie
Mysterious series: Sophie, Firis, Lydie & Suelle, Sophie 2 (to be released next February)
Secret series: Ryza 1, Ryza 2, ??? (not finished or announced yet)
It's generally recommended that you start with the first game in a series so that you can follow the same characters over multiple games as they grow and mature, character events and callbacks are a huge part of the games appeal because there's no real "main plot" (at least nothing dramatic of epic like in a typical JRPG, you're never saving the world or killing gods). Cutscenes and character events can compose almost half of a game's playtime.
Ryza 1 is probably the easiest to get into since it's so casual, while Sophie 1 is another good choice (most popular Atelier game in Japan ever) especially since it's getting a sequel in just a few months. Note that Sophie 2 is a direct sequel to Sophie, it takes place between Sophie and Firis, so you don't have to play FIris and Lydie & Suelle to be able to understand Sophie 2. (Although you should play them anyway, they're excellent games).
Arland, Dusk and Mysterious got re-releases called "DX", it's recommended to go for the DXs instead of the vanilla games, they have all the DLCs and even some unique never-seen before content, they're all available on PC.