I had already known for a long time that Ulfila's translation of the New Testament into Gothic was older than Jerome's Vulgate; but that chronological tidbit didn't really strike me until recently, and when it did, I thought it might be fun to chronologically list some early Bible translations.
My interest in this topic is mostly linguistic, while the interest of many or most people who have looked into it has been greatly or mostly theological. It's difficult for me to sort out the more authoritative Biblical scholars from the less authoritative, in part because there are so many of them, and unfortunately, the biased nuts do not helpfully affix labels in bold print at the head of their papers saying WARNING: BIASED NUT. DO NOT USE FINDINGS. Despite the lack of such labels, gross bias is often easy to spot, as when a member of a particular denomination affixes a significantly earlier date than anyone else to the translation most closely associated with his denomination, and acts as if he has never heard of the more conventional dating.
We do not know when some of these translations first appeared, and can only say that they are first firmly attested at such and such a date, and more weakly at such and such an early date, and speculate about the translation's beginnings.
Such is the case with the oldest-known (to me) Bible translation, the Septuagint, the translation of the Old Testament into Greek. The best I can do is to tell you that this translation was made in the 3rd and 2nd centuries in and/or around Alexandria, where a Jewish community had resided long enough that many or most of them were more familiar with the Greek language than with Hebrew.
Next come Syriac and Old Latin translations of the New Testament. (The term "Old Latin" is used to distinguish the earliest Latin Biblical translations from Jerome's Vulgate.) There is evidence of translation of parts of the New Testament into both of these languages both as early as the late 2nd century.
Next, we have evidence of translations of parts of the New Testament into Coptic going back as far as the 3rd century.
And then comes the "Gothic Bible," or to be more exact, the Gothic New Testament translation by Ulfila. It is well-established that Ulfilia (ca 311-383) was the translator.
Next, Jerome's Vulgate, begun after 382 and finished by 405. Recently scholars have been falling all over each other in the rush to proclaim that it is not correct to call this work Jerome's Vulgate, because not every single bit of the translation is Jerome's work, which is true, but most of it is by Jerome, and he at least inspected and approved the rest in the "Old Latin" versions, and revised those parts to some extent -- so I personally have no problem calling it Jerome's Vulgate. Just be warned, some people do have a problem with that.
It is with no great confidence at all that I guess that Biblical translations into Armenian, Ethiopic and Georgian began in the 5th century. I could be wrong, for all I know they could have begin earlier or later. There may be some really great and authoritative scholarship on the origins of all three of those written languages, but I haven't found any of it yet.
And finally there are Saints Cyril and Methodius, Byzantine missionaries to the Slavs. They are said to have translated parts of the Bible into Old Church Slavonic in the 860's. But some sources say they did this, while other sources say they "are credited" with doing this, which looks to me like scholar-speak for "they didn't do it, but for a long time a lot of people have thought they did, and I don't want to get into the middle of a huge argument right now." So I'm going to guess (guess!) that biblical translations into Old Church Slavonic began some time before 900, possibly by Cyril and/or Methodius.
By 900, writing in vernacular German had begun, and it would soon get underway in French and Spanish, which meant some translations into those languages of some parts of the Bible, although the Vulgate was well-established all over Western Europe and would remain overwhelmingly the preferred version there for centuries to come.
And if you're asking, Well, so what?! then I say: Well, it sort of lends a little bit of perspective to the 21st-century squabbles, in some English-speaking regions, over the 17th-century King James Version, doesn't it? and to the uproar caused by the KJV and Luther's German Bible and by the great unwashed in England and Germany learning to read at last. Every one of the translations of the Bible I've listed above, with the single exception of the Gothic version, has been continuously used by a wide reading public ever since it was first made.