The Federation typically did not make first contact with a species until that species had reached a certain level of technological advancement (usually the development of warp drive). The Prime Directive prohibited the contact with pre-warp species.
Human-Vulcan first contact
Humanity's first contact with the Vulcans was its first contact with an alien species. In 2063, the first successful test of the warp drive of Zefram Cochrane's Phoenix attached the attention of the Vulcan vessel T'Plana-Hath that was passing through the Sol system. The T'Plana-Hath tracked the Phoneix back to Bozeman, Arizona where it landed and made first contact. (Star Trek: First Contact)
Federation-Vendoth first contact
The first contact between the Federation and the Vendoth was made in 2376 by the USS Dragon-A. The USS Dragon had been accidentally transported to the distant Kalium Galaxy in 2377 and in the process arrived there in the year 2374. After two years of exploring this new galaxy the Dragon encountered the Vendoth. The Dragon interfered in the Vendoth's affairs, which resulted in the Vendoth launching an attack against Earth. The Vendoth's method of traversing from Kalium to the Milky Way resulted is several vessels from the fleet being deposited at various locations and times throughout the Milky Way. One group of vessels arrived in 2293 making the Federation aware of their existence nearly a century before the Dragon encountered the Vendoth. This event rippled through time and resulted in the Dragon's computer containing information about the Vendoth. (Star Trek: Unity)