Short Version: You can't go wrong with any of the main editions of the Vulgate, but I do recommend a version with at least notes or facing page translation (like the Dumbarton Oaks Edition). The Castellio translation is certainly interesting due to its Ciceronian language, but it has no notes or vocabulary, so it will definitely be more challenging for the student. I list some teaching-oriented editions of the Vulgate at the end of the post.
The oldest Latin versions of both old and new testaments, is the Vetus Latina, which is a collection of various works by different authors, mostly anonymous; as Tertullian wrote "Latinorum interpretum infinita varietas". These are of variable quality and hence not recommended to the student.
In the 380s AD Jerome wrote the Biblia Vulgata as a standardized Latin text of both testaments. The most common modern edition is the Clementine Vulgate of 1592, but there is also the Nova Vulgata published in 1979 by the Catholic Holy See that is intended to be a 20th-century improvement on the traditional vulgate of Jerome.
Sebastien Munster, a Swiss Calvinist, wrote a Latin version of the old testament which was published in 1532.
John Calvin himself wrote an edition of the new testament published in 1540.
Sebastian Castellio, a French protestant, wrote a Latin bible published first in 1550, the Biblia Sacra Ex Sebastiani Castellionis Interpretatione. This translation was written to be in classical Ciceronian Latin style as opposed to the vulgate, which was written in the ecclesiastical Latin typical of 4th-century Christians.
In 1569 at Geneva was published a protestant Latin old testament by Immanuel Tremellius and new testament by Theodore Beza. The Beza text had his Greek edition side by side with his Latin with copious notes and found final form in the 5th edition of 1598. This version influenced the Geneva Bible and King James Bible and was considered standard by protestants who could read Latin, like Milton and John Donne.
In general, most of the texts, both Catholic and Protestant, follow the style of the original Vulgate fairly closely and from the student's perspective are similar to each other. The Weber-Grayson is a standard edition with brief critical notes. One of the nice things about the Beza edition of 1598 is the facing Greek text with extensive notes. The Dumbarton Oaks Medieval Library (Harvard) edition of the Vulgate has an English translation on the facing page.
The Castellio bible is a different animal because of its Ciceronian style. Some Christians have criticized it because it does not use "standard" words for some theological terms and concepts. On the other hand, you are getting a somewhat more sophisticated and classical form of Latin prose. There are no notes in the Castellio. You just get the straight text only.
If you are still early in your Latin career I would suggest using a text with translation (like the Dumbarton Oaks editions), or at least notes. There are some specially designed editions that are specifically for students like the book "Vulgate Verses" which has the Latin text arranged by grammatical category and notated. There is a Gospel of John by Virginia Grinch that has Greek and Latin text with notes and vocabulary for students. A similar book is a heavily notated edition of Psalms by Davi Ladouceur. Wheelock's Latin Reader has a whole chapter devoted to Vulgate readings. Another didactic work is the Vulgate of Mark by Dale Grote, but note that it uses a modernized, simplified version of the Latin text for students. There are also purely lesson-oriented versions of Vulgate such as the Linguica Biblica Student which has selections from the Vulgate with vocabulary and exercises.
