Summary: No, we don't.
Hobbits are an unobtrusive but very ancient people, more numerous formerly than they are today;
The problem with this is that it's not clear if Hobbits are ancient from today's PoV or from the PoV of the time of LotR. The former seems more likely, especially since the rest of the paragraph deals a lot with Hobbits' interaction with the Big Folk today. So this tells us nothing.
Here's the money paragraph (my emphasis):
It is plain indeed that in spite of later estrangement Hobbits are relatives of ours: far nearer to us than Elves, or even than Dwarves. Of old they spoke the languages of Men, after their own fashion, and liked and disliked much the same things as Men did. But what exactly our relationship is can no longer be discovered. The beginning of Hobbits lies far back in the Elder Days that are now lost and forgotten. Only the Elves still preserve any records of that vanished time, and their traditions are concerned almost entirely with their own history, in which Men appear seldom and Hobbits are not mentioned at all. Yet it is clear that Hobbits had, in fact, lived quietly in Middle-earth for many long years before other folk became even aware of them. And the world being after all full of strange creatures beyond count, these little people seemed of very little importance.
So JRRT explicitly says that Hobbits are missing from Elven records and histories and those records are all that has survived from the First Age. Supporting that, there is no mention of Mannish records surviving from the First Age, and while the Numenoreans may have known of Hobbits, few of their Second Age records survive and none about Hobbits are mentioned.
Nor did Hobbits retain records (my emphasis):
Of their original home the Hobbits in Bilbo's time preserved no knowledge. A love of learning (other than genealogical lore) was far from general among them, but there remained still a few in the older families who studied their own books, and even gathered reports of old times and distant lands from Elves, Dwarves, and Men. Their own records began only after the settlement of the Shire, and their most ancient legends hardly looked further back than their Wandering Days [still Third Age]. It is clear, nonetheless, from these legends, and from the evidence of their peculiar words and customs, that like many other folk Hobbits had in the distant past moved westward. Their earliest tales seem to glimpse a time when they dwelt in the upper vales of Anduin, between the eaves of Greenwood the Great and the Misty Mountains. Why they later undertook the hard and perilous crossing of the mountains into Eriador is no longer certain. Their own accounts speak of the multiplying of Men in the land, and of a shadow that fell on the forest, so that it became darkened and its new name was Mirkwood.
For the Hobbits history -- still not recorded, mostly -- started only half-way through the Third Age:
About this time legend among the Hobbits first becomes history with a reckoning of years. For it was in the one thousand six hundred and first year of the Third Age that the Fallohide brothers, Marcho and Blanco, set out from Bree...
And that's what we know.