Babylon 5 has a number of first contacts not going as well as they could:
Earth has at least one truly disastrous first contact in its back story: The first contact of humans and Minbari happened between two squadrons of warships. Recognizing the strangers as fellow warriors, the Minbari kept their stealth at minimum and opened all gunports as a weird gesture of friendship (the message being "this is all our weapons, we aren't hiding anything"), spooking the EarthForce squadron, and when the Minbari flagship did charge its weapons to deal with a pest that EarthForce scanners couldn't detect the humans opened fire... And killed the Minbari's political and religious leader, who, anticipating the chance of this happening, had ordered to close the gunports right as his ship's skipper charged the weapons. The Earth-Minbari War saw the decimation of Earth's military and came dangerously close to the extinction of humanity. The Minbari fleet was literally minutes away from breaking through the final line of defense and being able to firestorm Earth when their leadership, the Grey Council, broadcast their own surrender and promptly retreated. Why they did this is one of the main mysteries of the early part of the series.
Earth got lucky in that they met the Centauri in their Second Republic, long after they abandoned their ancient expansionistic ways for the most part. Others met them in their First Republic and got invaded, with the Narn being invaded even if they met them around the same time as Earth due a combination of them being contacted by a different Centauri House, the Centauri needing the mineral resources, and the Narn being technologically much more primitive at the time.
The Narn first contact with aliens had happened when the Shadows built a base on their homeworld during their latest war with the Vorlon. The Narn fought back and managed to destroy their base... And lost all their telepaths in the Shadows' retaliation.
The Centauri's first contact happened in the age of the First Empire, when they were Actual Pacifists and reached the southern continent of their homeworld, finding it inhabitated by another sentient race, the primitive Xon. The Xon attacked for no apparent reason,note according to the Centauri. The Expanded Universe states that an alien posing as a god helped the start of Centauri civilization and decimated the Xon in the past, hinting that the Xon got scared when they saw the Centauri using their god's symbol and decided to strike first destroying the Centauri's First Empire... And getting completely wiped out by the First Republic that replaced it.
The Centauri's first (official) contact with people from outer space happened late during their war against the Xon, when the southern continent had for the most part fallen under their control and a trio of Technomages (considered renegades by the others) visited the planet and helped the Centauri progress technologically from an early Renaissance level to early 20th century level. The second contact happened when the Shroggen, minions of the Shadows, tracked those three Technomages down to Centauri Prime and, realizing they intended to make a stand using the natives as their army, attacked without bothering to explain the situation. After the Shroggen were beaten back and the Technomages had left, the Centauri swore to reach space and form a mighty nation to protect the weak from those who would subjugate or kill them, the first step into becoming a conquering empire.
One episode deals with a probe that gives the people on the station a short amount of time to answer various complex mathematical and scientific questions before it blows up, as its creators believe only races that can answer them deserve to live. However, Sheridan figures out it's actually designed to kill anyone who can answer the questions, whom the aliens consider a possible threat.
Dark Skies also has the first contact between the Hive and the American government take place in secret. The President has a face-to-face meeting with an alien representative, who telepathically demands Earth's unconditional surrender. In response, the President creates the Majestic 12, whose job is to cover up the existence of aliens and fight their attempts to infiltrate humanity using any means necessary.
Features in the backstory of Defiance. It went... badly. The Votan arrived as desperate refugees, and relations between them and us were always strained, and then one of their ambassadors was assassinated on live TV. The resulting wars reduced the planet to mostly 19th century infrastructure levels (give or take some alien gizmos).
Played with. In "I, E.T.", Moya has to land on a backwater planet because of a beacon put on her that could alert the Peacekeepers, and removing the beacon is a surgical procedure that could kill her. John and the others go in search of an element that can serve as an anesthetic for the procedure and meets with some of the locals... who are flabbergasted at meeting an honest-to-goodness alien. Yes, a human is the first contact of another sentient species.
Also happens on Earth multiple times throughout the show, though the first few times it was All Just a Dream. Eventually it happens for real, though.
One memorable episode, "Trial by Fire", starts with a new US President being inaugurated only to be whisked away into a bunker deep underground when objects are detected approaching Earth. There's plenty of misunderstanding, as the humans have no idea what the aliens are planning and cannot understand the language in the messages they send. Meanwhile, the President's efforts to maintain peace are threatened by Russia's gung-ho attitude to the aliens, eventually forcing the President's hand and causing them to launch nuclear weapons at the aliens, to no avail. The bitter irony is revealed at the end, just before Washington, D.C., and Moscow are obliterated in retaliation. The aliens were speaking English all along, but their message was distorted by their aquatic environment. It was a message of peace.
Another episode, "Relativity Theory", involves a human ship landing on an Earth-like world and encountering a group of seemingly primitive aliens. The scientists are keen to make contact and study the locals, while the military types want to wipe out the locals to make room for a future colony. It turns out the "primitive locals" were the equivalent of Boy Scouts on a camping trip and weren't even from that world. When they call for help, a powerful alien ship shows up, downloads the human ship's navigational charts, destroys the human ship, and heads straight for Earth to exact revenge.
Notably averted in Power Rangers, where hostile aliens attack on a weekly basis for six years (without any diplomatic contact of any kind!) before anyone friendly enough (and not focused on perpetuating The Masquerade) to speak to officials or the public at large shows up. By that point, everyone is fairly certain that there's life on other planets; it's the reason property values in Angel Grove have fallen so much recently.
Vaguely mentioned in an episode of Sliders, after they end up in a world with a more advanced level of technology. It turns out, due to World War II continuing for several more years in this world, a different US President got elected just in time for the Roswell incident. Instead of covering it up, this President decided to make the existence of aliens public knowledge. Shortly after that, the Reticulan-American Free Trade Agreement (RAFTA) is drafted, allowing a good amount of Imported Alien Phlebotinum to be ubiquitous by the end of the 20th century, as well as a manned mission to Mars. No aliens are actually present in the episode, but one of the locals is a human who has been accidentally turned into a Half-Human Hybrid with a gene therapy that cures most known diseases.
The real First Contact in the Stargate-verse (not counting Transplanted Humans abducted thousands of years ago, or the Precursors who happened to evolve here millions of years ago (sort of, maybe)) happened in 1994, when a top-secret Air Force team used an ancient alien artifact to travel to another world, where they found Transplanted Humans being ruled by a Puppeteer Parasite. More than 15 years, 300 episodes, three TV series and two made-for-TV movies later, despite the creation on Earth of half a dozen ships capable of interstellar travel and at least two battles in Earth's atmosphere or in orbit between humans and aliens, the existence of aliens is still apparently a secret from the general public. This trope has also been inverted several times, when we see First Contact from an alien (well, transplanted human) point of view. In general, the aliens' government covers it up just like ours if they are advanced enough to do so, but there have been several exceptions.
Ironically for a show revolving around specialists in alien diplomacy, the Enterprise-D was pretty bad at it. Their crewmen were captured, their surveillance apparatuses blown wide open, their technology pillaged. When the Federation is probed by a heretofore unknown race (and not the other way around), things typically go much smoother.
The self-titled Next Gen episode "First Contact" depicts a First Contact gone awry. The species in question erases all evidence of their encounter with Picard's crew, believing that divulging the existence of aliens would cause panic and civil unrest. Even some warp-capable species want nothing to do with those weirdos.
"Who Watches the Watchers" had the opposite effect Starfleet intended, with Picard accidentally rekindling the Minatakans' belief in a wrathful god.
Sometimes when First Contact occurs between two alien species, the results have been less than happy. The Cardassian Union landed on Bajor and proceeded to herd everyone into ore mines. They later claimed they were a backward race and disputed that Bajor had achieved space flight first (they had).
In Enterprise's Mirror Universe episodes, the First Contact between humans and Vulcans at first played out exactly as in the film, but instead of welcoming the aliens with a handshake, the humans shot the aliens, stole the advanced technologies on board the alien ship and proceeded to build an interstellar Terran Empire.
Another ENT episode reveals that Vulcans have observed Earth before the official first contact. A trio of Vulcans crash-landed in 1957 Pennsylvania, with the captain dying. The two survivors, including T'Pol's great-grandmother T'Mir, managed to blend in for months before a rescue ship arrived. However, one became enamored with human culture and chose to stay behind, with T'Mir telling the rescuers that he died as well.
PIC reveals that a young boy was searching for his dog in the woods in the late 20th century and ran into two Vulcans running scans. He ran away but the Vulcans caught up to him. One of them tried to mind meld with him in order to erase that memory, but they were beamed away before he completed the process. The boy became obsessed with aliens and joined the FBI when he grew up. When Picard and his group ended up in the past, the agent tracked them down and arrested Picard and a young Guinan.
An episode of Time Trax mentions that a humanoid race called Procardians have made radio contact with Earth in the 22nd century, and their diplomatic delegation is on the way. Except Darien learns that Procardians have visited Earth in the late 20th century, twice. The first time, the research team found humans wanting and left. However, a girl remained behind. A few years later, her lover stole a ship and returned for her. Darien helps him find her, but since they don't have a ship to return home, he offers to send them to his own time to await the arrival of the delegation.
The very beginning of V (both the original series and the remake). Alien ships appear over Earth, and the humanlike aliens promise that they have come in peace and want to establish relations with mankind. Their motives and nature turn out to be far more sinister.
In The X-Files, humans were contacted, abducted, and experimented upon for millions of years but the first contact between the alien Colonists and human authorities took place, as far as we know, in October 13, 1973, giving rise to the shadowy Government Conspiracy to hide the truth. It is also of note that the contact was made possible by extensive reverse-engineering of the alien craft recovered from the Roswell Incident.
