We've never really been told "Species X achieved warp in year Y". Instead, we've been shown that humans on Earth first achieved (manned?) warp drive in 2063, and that numerous species had warp drive in the 2150s when the show ENT took place.
Major or familiar species that had warp in ENT:
Klingons (no mention how long they had had it)
Vulcans (again no mention, but a ship of theirs had a "warp engineer" aboard in 1957 already)
Romulans (again no mention)
Andorians (again no mention)
Tellarites (again no mention)
Ferengi (again no mention)
Borg (again... you get the picture)
Now, each of these species seemed "well established" in interstellar affairs, so presumably they had been warping for several decades at the very least. But we know that Vulcans had been interstellar for 3,000 years at that point, or that they at least had a 3,000-year-old temple in another star system at that point. We also know that Romulans reached their home planet at least 2,000 years before ENT, but whether they did so with warp, impulse, or just flapping their ears real hard, was never told. We further know that Klingon history includes references to starflight: Kahless himself said Klingons would meet him again on a distant star, a millennium before ENT. But we don't know the exact timetable of the early Klingon spatial exploits, and we don't know whether they were being poetic or literal with some of their spaceflight accounts.
The Ferengi spent 10,000 years struggling towards warp drive, as Quark laments in "Little Green Men", but there's no data on when they achieved their goal (that is, managed to buy the secret from an unknown seller). Quark also thought he could become that seller when time-traveling back to 1947, and claimed that the Ferengi would then get warp "centuries before humans or Klingons or even Vulcans". But of course, 1947 would not be centuries before 1957, so Quark was either in error there, or then simply thinking that he should become a time-traveling salesman now that he knew how.
The Borg used warp in ENT, but we don't know if this was their first warp experience ever (time loops notwithstanding), or perhaps the billionth anniversary of their first flight.
Despite decades upon decades of Star Trek, then, the writers have managed to keep things delightfully vague. We may postulate that Vulcans had warp several thousand years ago and then lost it in wars, or that they invented it in 1934. We may say that Klingons during Kahless were a starfaring empire already, or that they first learned to warp in 2098.