So, given that the Ys games have seen so many remakes and reimaginings, it's hard to see in which order the games should be played properly. Of course, if you're going the exclusively "legal" route, you have few options but to obtain a PlayStation Vita and forego Ys 5 entirely, but as I see it, there are different ways to play this series:

(Note: Where the game is available on PSP and through PSN, it stands to reason that it should be playable on Vita)

The "hardcore traditionalist" approach (playing only the earliest "original" versions, usually requiring Japanese skills or an unofficial translation):
Ys 1: Ancient Ys Vanished (PC-8801)
Ys 2: Ancient Ys Vanished (PC-8801)
Ys 3: Wanderers from Ys (NEC PC-9801, MSX2, Sharp X68000)
Ys 4: Mask of the Sun (SNES) OR Ys 4: The Dawn of Ys (TurboGrafx-16)
Ys 5: Lost Kefin, Kingdom of Sand (SNES)
Ys 6: The Ark of Napishtim (PC, PS2, PSP)
Ys Origins (PC, PS4, Vita)
Ys Seven (PC, PSP)

The "mildly traditionalist" approach (playing only the earliest English-language "original" versions):
Ys 1: Ancient Ys Vanished (Master System, DOS, NES)
Ys 2 (TurboGrafx-16)
Ys 3: Wanderers from Ys (Sega Mega Drive, SNES, TurboGrafx-16)
Ys 4: Mask of the Sun (SNES with unofficial translation patch) OR Ys 4: The Dawn of Ys (TurboGrafx-16 with unofficial translation patch)
Ys 5: Lost Kefin, Kingdom of Sand (SNES with unofficial translation patch)
Ys 6: The Ark of Napishtim (PC, PS2, PSP)
Ys Origins (PC, PS4, Vita)
Ys Seven (PC, PSP)

The "semi-modernist" approach (playing the most recent canonical versions, where possible with official translations):
Ys 1 Chronicles (PC, PSP, Android, iOS)
Ys 2 Chronicles (PC, PSP, Android, iOS)
Ys 3: The Oath in Felghana (PC, PSP)
Ys 4: Memories of Celceta (Vita, probably PC at some point)
Ys 5: Lost Kefin, Kingdom of Sand (SNES with unofficial translation patch)
Ys 6: The Ark of Napishtim (PC, PS2, PSP)
Ys Origins (PC, PS4, Vita)
Ys Seven (PC, PSP)

The "absolute modernist" approach (playing the most recent canonical version, where possible with official translations, according to narrative chronology)
Ys Origins (PC, PS4, Vita)
Ys 1 Chronicles (PC, PSP, Android, iOS)
Ys 2 Chronicles (PC, PSP, Android, iOS)
Ys 3: The Oath in Felghana (PC, PSP)
Ys 4: Memories of Celceta (Vita, probably PC at some point)
Ys 5: Lost Kefin, Kingdom of Sand (SNES with unofficial translation patch)
Ys 6: The Ark of Napishtim (PC, PS2, PSP)
Ys Seven (PC, PSP)

Or Origin, then Ys I & II, then continue with whichever you want. Other than the Ys trilogy the other games has no connection with each other, and just by the in-game informations there is no cronology included. Pretty much completelly separate titles that does not shares anything with any other games.

I like all these games and still have the original discs, however only Origin and I & II have a story, the others have only a plot. I have to accept the fact: the games are equally good but only the trilogy is legendary.