There are two recent trilogies, the Arland trilogy (Rorona, Totori and Meruru) and the Dusk Trilogy (Ayesha, Escha & Logy, Shallie). Within trilogies, there are similarities in e.g. alchemy systems and general progression, but each game has its own unique take on it:
- Rorona introduced the basic formula. Note that it has had a remake which is preferable in all regards to the original.
- Totori is one of the more personal-story-focused, and has a bit more of an adventure/coming-of-age feeling
- Meruru has a system where you (the princess) are tasked with rebuilding and improving your Kingdom
- Ayesha I have no idea about because it was released on PS3 without dual audio so I was waiting for the Vita release
- Escha & Logy has two protagonists to select from.
- Shallie, as far as I know, will change the main goal accomplishment system away from the current time management concept and introduce free third person camera control (?)
Within the individual trilogies, there are some recurring characters, so ideally those should be played in order. There's no significant connection between trilogies.
As for which one to start with, in my opinion the best one of the 6 PS3-era games is still Totori, but I guess the new Rorona is also a good place to start. Perhaps Ayesha is too.