I don't think I've given much thought to how humanity handles First Contact, so here's some brainstorming:
This is how it plays out for Bob and the other hams in the park with him the night of First Contact:
-Bob attempts to make a contact through a recently launched amateur satellite (What hams call "squirting the bird").
-He tunes into the sat's beacon signal, which is just it's call sign repeated in Morse. Right after tuning in, the signal abruptly stops.
-Bob chalks it up to someone else hogging the transponder output power and starts sending his call sign and asking for a response from any listening station ("Calling CQ").
-After a minute or so of repeated CQ's he gets a response, which is just "CQ CQ CQ CQ CQ CQ" spammed over and over again.
-Bob responds with something along the lines of "You must be a new ham. Happy to see you working satellites, but this isn't how a contact is supposed to work." (I'm expanding what would in reality be a heavily abbreviated string of letters).
Here's how a CW QSO over satellite is supposed to go.
-The mysterious other party spams CQ CQ CQ a few more times and then stops.
-Bob goes to bed.
-He wakes up well before sunrise to go to the bathroom but can't go back to sleep, so he exits the camper/tent/etc and just stands outside staring at the stars.
-Quick note about Bob. Like a lot of hams he's an odd fusion of geek and redneck. He was raised on his father's pulp sci-fi collection and golden- and silver-age comic books, and has the typical interest in tinkering common to all hams. He is also the typical friendly but uncomplicated and blunt sort of person you'd associate with semi-rural Texas. In short, he's a genre-savvy redneck.
-He sees a UFO land in a nearby clearing. It drifts to earth under a parachute, having dropped like a stone from orbit rather than landing like a shuttle. It's roughly the size and shape of a camper van.
-He stares dumbstruck at it, fearing to come closer. This state of affairs continues until morning twilight.
-He hears stirring from within the craft, and cautiously approaches, crouching behind some bushes to remain hidden.
-As the sun breaches the horizon, a hatch noiselessly opens and a white-furred creature the size of a large dog emerges, looks around, then re-enters the craft.
-Bob hears quiet yipping and growling coming from inside the craft, then the same creature re-emerges along with five others, one of whom is completely hairless. A rod surmounted by a decorated metal ball is wrapped in the white one's tail. The creatures look like monkeys with the head of a fox, and thus Bob mentally starts calling them "monkey foxes" in want of a proper name.
-The nekkid monkey fox starts looking over the other five, uttering louder yips and growls, which Bob has now deduced is their language.
-As this point Bob knows that aliens have landed, that there are only six of them, and that they do not appear to be making any effort to stay hidden. They appear to be unarmed (the rod born by the white-furred creature does not seem to be a weapon), so Bob cautiously emerges from the bushes.
-The aliens notice him approach, and all but the first white-furred one retreats back to the ship. It barks a few times in their direction, then turns to face him.
-It backs up a few paces, its ears pinned back, then it unceremoniously pukes a bunch of translucent yellow goop onto the grass.
-It shakes itself off in canine fashion, dropping the rod in the process. The metal ball at the top bursts open, and bluish white milky liquid spills out, soaking into the dry brown grass. The creature glances back at the mess, then back at bob, an unreadable expression painting its vulpine countenance.
-Bob notices the creature is craning its neck to meet his gaze, so he sits down in the grass to make it more comfortable.
-This seems to embolden the creature and it comes closer. It barks a few more times in the direction of the craft, and the others tentatively re-emerge.
-The creature rears up on its hind feet, pats its belly twice with its left front extremity, then sits back down. It repeats this gesture again, and bob mimics it, assuming it to be a greeting.
-The creature utters something, gesturing downward at its feet with its muzzle. It utters the same thing while pointing its muzzle at its conspecifics behind it. As with the greeting, it repeats this a few times. Bob catches the following repeated /yip, whine, huff, growl, huff/.
-Bob attempts to repeat the utterance, guessing its their name for themselves, but all his human vocal tract can manage after a few tries is “yinrih, yinrih”.
-Bob pats his chest and says “Human, human”. After a few repetitions, the creature manages /huff, grunt, huff, grunt/.
-At this point, Bob's fellow hams have approached, standing a ways behind him and the aliens. They express fear and skepticism about this encounter, suggesting that the authorities be called.
-Bob, as I said before, is genre savvy, and there’s only two flavors of alien encounters he’s aware of: Either they come in peace or they don’t. If they do come in peace, the government captures them and cuts them up. They’ve shown no hostility up to this point, and besides that, they’re fuzzy and cute, and it’s always the adorable ones that end up on the dissection table. Bob delivers an impassioned speech along the lines of “This is America, they’re innocent until proven guilty. If they are hostile, then we shouldn’t be the ones to fire the first shot, Texas means ‘friend’, and we should greet them as friends,” etc.
-Bob turns back to the white-furred one who he assumes is the leader and extends his hand to offer a handshake. It looks at his outstretched limb, its head tilted in puzzled bewilderment.
-Bob attempts to demonstrate, clasping his hands together and pumping them in a ‘one man handshake.’
-The alien rears up and clasps its own forepaws together and pumps them in exact imitation of Bob’s gesture. This elicits a chuckle from Bob, but he persists in demonstrating the correct way. He pulls one of the other humans forward and demonstrates a proper handshake with this second party, repeating several times.
-He faces the alien and extends his hand again. It reaches forward slowly. Bob notices the features of its forepaw: thick doglike pads, sharp red claws, six digits (inner thumb, four fingers, and outer thumb). The claw next to its inner thumb is flatter and broader than the others. Hand and paw finally grasp one another, and First Contact is achieved.