Ys is one of my favorite series, so I'll provide all of the info I can. (Which is far more than you ever wanted to know.)
Ys I and II:
Many iterations of these two have been released, more than even FF4. The Chronicles version (for PSP and PC) is the best in pretty much every respect. That said, I and II are still very much products of their time, so don't expect them to go as smoothly as the later games. For instance, there'll be a lot of instances where you'll only be given vague instruction on where you'll need to go next, or you might pass right by an item that'll be required later, without ever being informed of its existence. They also have a rather unusual combat system in which you attack enemies simply by running into them. The catch is that if you run into them head-on, you'll be hurt instead, so you need to attack them off-center or from the sides or behind. It's quite fun once you get used to it. Overall, I and II definitely still worth playing if you don't mind some dated design here and there. (Also, I think this
was the song you were mentioning. This
is the Chronicles version. Or it might have been Termination.)
Ys III: Wanderers from Ys:
The original version of Ys III (PCE/TurboGrafx and virtually everything else in that era) was sidescrolling, somewhat like Zelda II. It controls rather poorly, at least compared to the other Ys games, and is generally considered to be very difficult. It received an excellent, highly polished remake (PC and PSP) called Ys: The Oath in Felghana. Felghana completely overhauled and reimagined III using the Napishtim engine, and introduced the double jump (more on the Napishtim engine later). It remains arguably the most difficult game in the series, however.
There are actually four different Ys IV's, only one of which was actually developed by Falcom. The four share many common elements, but differ in many areas as well. Memories of Celceta is also the only one of the four to see an official English release.
Ys IV: Mask of the Sun (SNES) was Tonkinhouse's version. It features the same bump system as I and II, but is more complex, in both gameplay and story.
Ys IV: Dawn of Ys (PCE/TurboGrafx) was Hudson Soft's version. It is generally considered to be better than Mask of the Sun, due to having 8-directional movement (a first!) and much better sound quality. The story, while sharing many characters and plot elements, also takes some different turns, and has a different main villain.
Ys IV: Mask of the Sun: A New Theory (PS2) was Taito's remake of Mask of the Sun. It plays more like the Napishtim-engine games (again, we'll get to that,) and takes some of its own liberties with the story. Generally not considered to be that great.
Ys IV: Memories of Celceta (Vita) is Falcom's own version. It has gameplay similar to that of Ys Seven (more on this later,) with a few new additions. It also has a huge overworld that is mapped as it is explored. As usual, the story is a bit different from the other iterations. This is now the canonical version. (Mask of the Sun was previously.)
Ys V: Lost Kefin, Kingdom of Sand (SNES) (not official title):
The only Ys game to not receive an official English release in any form (yet, at least). It's something of a black sheep, featuring much slower gameplay than the other games and being somewhat oddly balanced. It also uses a completely different musical style from the rest of the series, with a somewhat more orchestral sound. It received an "upgraded" PS2 port from Taito, which changed a number of systems, not necessarily for the better. Lastly, and most importantly, it was the first game in which Adol is able to swing his sword, rather than using the bump system.
Ys VI: The Ark of Napishtim (PC/PS2/PSP):
This game was the first 3D entry in the series. It features such new abilities as jumping, dash jumping, and the ability to use multiple different magic swords with their own unique abilities (which would take on various forms in the later games, but the concept would remain similar.)
Ys Origin (PC):
Origin is a prequel to Ys I and II (by 700 years). It's very linear and intensely focused compared to the other games in the series, and is the only one not to star Adol Christin as the protagonist. Instead, there are multiple other characters with differing fighting styles. It uses a more refined version of the Napishtim engine, building on the original features and those added in Oath in Felghana.
Ys SEVEN (PC/PSP):
SEVEN (officially stylized that way) introduces a new engine, and brings with it some changes. Jumping has been removed again in favor of rolling/dodging. Seven also is the first in the series to utilize a party system, allowing for switching between three selected members on the fly (out of an eventual max of seven). They all have their own unique equipment, fighting style and skills.
(On a side note, the way "Ys" is stylized in Seven's logo makes me cringe. It's like Falcom wants to perpetuate the idea that there's an apostrophe.)
The stories, while not generally amazing on their own, do their job well enough and don't get in the way of the action. Ys is not known for its focus on story. That said, they do connect to each other and reveal bits of overarching meta-plots. The chronology is Origin > I > II > Celceta > Felghana > Kefin > Napishtim > Seven.
Ys is also well-known for its amazing music. A few of my favorite songs:ErnstValestein CastleSealed TimeTo Make The End of BattlePalace of DestructionA Great OrdealFieldRoad to Solitary DeathCrossing Rage!Scars of the Divine WingUnderground Ruin
Don't even get me started on Falcom's other big series, Legend of Heroes.