Kazuya Minekura's Saiyuki is a fabulous manga, but due to publisher switches and long hiatuses, it has never quite made it to the world of animation without some major confusion.
The manga is composed of four series: Saiyuki, Saiyuki Reload (which continues from the end of Saiyuki), Saiyuki Reload Blast (which continues from the end of Saiyuki Reload), and a Prequel series called Saiyuki Gaiden which stars the previous incarnations of the series' main characters.
Confused yet? We're only getting started.
If you are coming to the Saiyuki anime for the first time, the place you'll want to stay is Gensoumaden Saiyuki (or just Saiyuki), the original anime. It is based on Saiyuki (the manga) and includes roughly the first half of Saiyuki Gaiden.
The next anime series, chronologically, are Saiyuki Reload and then Saiyuki Reload Gunlock, which cover up to the middle of the Saiyuki Reload manga.
These OVAs you're currently on are just the ending of Saiyuki Gaiden. (Likewise, the Saiyuki Bural OVAs are simply a manga arc that was not included in any of the anime series but was a fan favorite so I guess it got OVAs.)
Basically, tldr, if this is your first taste of Saiyuki in any form, you're gonna be very, very confused. These OVAs are mostly for established fans of the manga.
That said, as an established fan of the manga, I've never been satisfied with the anime adaptations. For one thing, as you can see, they're overly confusing and brimming with filler. I also have noticed that Minekura's art seems difficult to adapt to anime--the animation is horrendous at worst, vaguely weird at best. Saiyuki Gaiden is some of the best animation Saiyuki has gotten, but it still looks kind of off, as if her anatomy isn't quite made for moving. Not my favorite.
That said, I do recommend the manga whole-heartedly.
