Taken from my self-made chonology - here's what happened between First Contact and 2079. I have listed only the relevant entries:
2064
Zefram Cochrane gives a commencement speech to the graduating class at Princeton. During his address on First Contact, his speech wanders into bizarre details of cybernetic creatures who tried to prevent his first warp flight and enslave Earth but they were defeated by a group of humans from the future. Known for his many tall tales and frequent intoxication, no one takes Cochrane seriously. A few years later he recants this statement.
("Regeneration" [ENT]. 89 years before the episode.)
2065
New Berlin, humanity’s first permanent lunar colony, is established on the moon. By the 24th century, New Berlin would become only one of many settlements on Luna which would be home to over 50 million people residing in atmospheric domes.
(“Terra Nova” [ENT], “Valiant” [DS9] and Star Trek: First Contact”. Date is approximate. Captain Archer noted that New Berlin was settled before the launch of the S.S. Conestoga in 2069, but sometime after Cochrane’s warp flight. It would be reasonable to assume the colony was established before manned deep-space exploration such as the Valiant and the Terra Nova missions.)
The S.S. Valiant embarks on a deep space exploration mission. The expedition is lost when its impulse engines cannot withstand a magnetic storm and the vessel is eventually swept out of the galaxy. The ship is nearly destroyed while trying to return across the energy barrier at the edge of the galaxy. Six crew members are killed, but a seventh experiences a mutation that dramatically enhances latent ESP powers, making that person a threat to the ship and its crew. The Valiant's captain later orders the ship destroyed to prevent the mutated crew member from escaping.
("Where No Man Has Gone Before" [TOS]. The Valiant expedition embarked 200 years prior to the episode.)
2067
British astronomer John Burke of the Royal Academy maps the area of space which includes Sherman's Planet. This region is later the subject of territorial dispute between the Federation and the Klingon Empire.
("The Trouble with Tribbles" [TOS]. Spock and Chekov's discussion established the date as 200 years prior to the episode.)
Earth launches "Friendship 1", a long-range probe designed to pave the way for manned spaceflight by reaching out to make contact with other species. It was densely programmed with translation matrices, scientific and cultural databases, computer chip designs, and transceiver instructions.
("Friendship One" [VOY]. Chakotay establishes the probe was launched before Starfleet existed. However, the markings on its nacelle show the familiar arrow-head design used throughout the various Trek incarnations which would indicate that the United Earth Space Probe Agency (UESPA) was using this design years before Starfleet was established. UESPA was first mentioned in "Charlie X" [TOS] in the Original Series. Later during the "Enterprise" series, we indeed see the arrow-head design used by the UESPA during the conference for the Coalition of Planets in "Demons" [ENT].)
2068
By this year at the latest, the Y-500 Class freighters are retired from service in space.
("Strange New World" [ENT]. Extrapolated from Mayweather's ghost story and Trip's comments.)
The Interstat communications radio code ceases to be used. Around this same time a transport carrying colonist from Earth becomes lost. They come to settle on the lone planet in the Cepheus system which they name after their mission name, Terra Ten; themselves being the tenth in the series of colony flights. However, the colonists are exposed to the native spiroid epsilon waves which shrink their molecular structure until they become 1/16 of an inch tall. The "terratins" develop their own unique culture, which they adapt to their small size which over generations becomes a genetic characteristic.
(Two centuries before "The Terratin Incident" [TAS].)
The Draco lizard, a small reptile native to planet Earth, becomes extinct.
("New Ground" [TNG]. According to Enterprise-D teacher Mrs. Kyle, this occurred approximately 300 years prior to the episode.)
2069
After establishing several colonies within the solar system, Humans dare to venture into the cosmos and plant a settlement upon another planet. After finding a habitable Earth-like world less than 20 light years away they launch the S.S. Conestoga under Captain Mitchell to undergo the nine year voyage in what was later dubbed as the "Great Experiment".
("Terra Nova" [ENT]. Date is deduced by Archer's comments.)
The fourth in a series on minor wars between Earth and the felinoid Kzinti comes to an end. In each case Kzin was always the aggressor but Earth managed to triumph over them each time because of their superior technology. After their final loss, Kzinti sign the treaty of Sirius whereupon their territories are demilitarized and they are forbidden to possess phase energy weapons.
(According to Sulu, 200 years before "The Slaver Weapon" [TAS]. Since this episode was based on novels of Larry Niven, we might consider the Earth-Kzin wars to be apocryphal to the Star Trek universe. According to Niven's "Known Space", the first war was fought with sublight vessels and the Kzinti could only be defeated because of Earth acquiring faster-than-light travel. It is interesting to note that had "Star Trek: Enterprise" gone into a fifth season, producer Manny Coto was planning on featuring the Kzinti and based upon a script edited by D.C. Fontana and Andre Bromanis. A conjectural design for a Kzinti spaceship (pictured) was even commissioned by writer Jimmy Diggs. This surely would have lent a better understanding to the awkward placement of the Earth-Kzin conflict which seems to predate the Earth-Romulan war.)
2073
Zefram Cochrane delivers the noteworthy quote, "Don't try to be a great man, just be a man. And let history make its own judgments."
("Star Trek VII: First Contact". Ten years after his warp flight.)
2076
En route to her new home, Vera Fuller, a colonist aboard the S.S. Conestoga gives birth to a daughter whom she names Bernadette.
(“Terra Nova” [ENT]. Records recovered by Enterprise crewmembers in 2151 determined that Bernadette was born about 75 years prior. She would become one of the children who survived the radiation fallout when Terra Nova is struck by an asteroid. Living underground, she would forget her human heritage and go by the name Nadet.)
2078
July 23
After nine years, the S.S. Conestoga finally arrives at their target planet which the colonists named Terra Nova – “New Earth”. After Captain Mitchell conducts a groundbreaking ceremony, the colonists disassemble the Conestoga and build a home for themselves out of its hull.
("Terra Nova" [ENT]. Trip noted that his grandfather, Charles Tucker the first, saw the first transmissions from Terra Nova when he was a child.)
2079
Although great strides have been made in space travel, Earth continues the difficult recovery from the Third World War. In this post-atomic horror, the legal system is a mockery, based on the principle of "guilty until proven innocent". True justice falls into disfavor as Shakespeare's suggestion to "kill all the lawyers" has been taken quite seriously.
("Encounter at Farpoint" [TNG]. Q says that this is the year that his courtroom re-creation was based on.)
As we see, not much has been revealed about Earth's socio- or political condition after WWIII up unto the so-called "post-atomic horror" depicted in "Encounter at Farpoint". What is apparent is that humanity's desire to explore space seems to move at full throttle almost from the start; even begining to colonize the moon at at quick pace. If I had to hypothesize, Q's court may have been indicative of conditions in the East, considering the Mandarian look of the prosecutor, the use of the gong, etc. Perhaps the West and the more industrial nations had an easier recovery after the War with enough commonality among the surviving nations to pool their resources together to revive a space program. In "Broken Bow", Trip tells T'Pol that humanity cured poverty and hunger within two generations. Considering that the court scene was only 16 years after WWIII, there were certainly areas on Earth that moved consideribly slower than others when it came to establishing "utopia".
