Babylon 5 as a series is actually quite a good example of how first contact scenarios might play out.
There was the Lieutenant Warren Keffer, a Starfury fighter pilot who saw a Shadow Battlecrab vessel in hyperspace once, then went looking for it and, due to his dangerous curiosity, attracted quite an unwelcome amount of attention from it in the process... but managed to record his encounter and launch a probe with the data back to his home station.
There was the historic and tragic moment of the first encounter between humans and Minbari, where a series of misunderstandings led to an overreaction of a ship captain with an all too itchy trigger finger, getting the the spiritual leader of the Minbari killed by a (un?)lucky critical hit in the process, enraging the Minbari council so much that they started a holy war that almost culminated into the total annihilation of the human race.
There was the infamous star system of Sigma 957, where strange sightings had occurred and explorers were warned not to enter. When Catherine Sakai was trying to make a survey sweep of one of the planets there, a gigantic, strange alien vessel appeared out of another dimension, floating there for a few seconds not doing anything to Catherine's ship... but as it returned to where it had come from, an energy spike caused the systems of Catherine's ship to overload and it's power core to malfunction, leaving it in an unstable descending orbit...
There was Susan Ivanova's attempt to gather all remaining "First Ones", the remainders of the oldest space faring races in the galaxy, trying to rally them against the Shadows and the Vorlons, and the delicate diplomacy she had to employ because every one of their ships was infinitely superior to her own one, and she could have been destroyed in the blink of an eye.
