As the Good Book Says...: Grandis said the following about Nadia wanting to work on the Nautilus: "if a man won't work, neither shall he eat."
Black Comedy Animal Cruelty: King's taste buds getting scorched by Grandis' cooking (and then the subsequent chase through the Nautilus).
Both Sides Have a Point: The Senior Staff's debate about taking on the children and the Grandis Gang as apprentice crewmen. Even without her personal bias and subtext, Electra feels their war with Gargoyle has reached a crucial stage and that they can't afford to blithely take on new personnel (let alone children) who could disrupt the carefully-crafted equilibrium onboard. The Chief Engineer's counterargument is that they're short on personnel and could use the extra hands. Furthermore, Jean, Nadia, and the Gang have all demonstrated they're reliable assets to the ship and have a mutual enemy in Gargoyle.
Critical Staffing Shortage: A plot point raised by the Chief Engineer during the debate. Since they're fighting a secret war against Neo-Atlantis, the Nautilus can't exactly dock at the nearest port-of-call and interview potential applicants to replace any crewmen lost in the conflict, can they? So at this point, the ship's not fully crewed and this ends up being a factor in the majority of the Senior Staff overruling Electra's objections to make them apprentice crewmen. Of course, Nemo has other reasons for keeping Nadia aboard...
Defrosting Ice King: Nemo after his first extended on-one-one conversation with Jean. While he remains aloof, he also becomes warmer and friendlier with Jean from this point forward. Given later developments in the series, in retrospect it's clear that getting to know Jean and seeing how much he cares about Nadia settles any doubts Nemo may have had about the boy in the previous episode.
Failed a Spot Check: Played for black comedy. King's so intent on stalking Grandis and stealing her fish that he doesn't pay attention to how much spicy seasoning Grandis is dumping onto the meal.
Electra objects to letting the Grandis Gang and the children remain aboard (and cites logistical and security reasons). While Nemo thinks part of it's jealousy over Grandis (which Electra denies), subsequent episodes will make it clear it's really Nadia that's the driving factor.
During the debate about the new recruits, it's mentioned that no one aboard the Nautilus fully comprehends its super science. They've learned how to use it, but they'd never be able to duplicate it on their own. This sets up the reveal later in the series that Nemo's crew didn't actually build the Nautilus. It's actually an Atlantean craft they broke out of mothballs and refitted for oceanic use against the Neo-Atlanteans. As Tarteoss refugees, Nemo and company have learned how to use the advanced technology, but duplicating it is well beyond 19th Century capabilities.
The Nautilus engine is most definitely not a conventional 19th century engine and doesn't look human manufactured. Hell, Nemo practically tips his hand about the show's Ancient Astronauts twist during his discussion with Jean in Engineering — especially how the ship's science is well beyond the 19th and even 20th century's scientific development.
Nadia stating she's earned her life since she was small, even though she's a teenager. This builds on her interaction with the Ringmaster in the Pilot, giving more clues about her early life in the Circus (and which will be confirmed in flashbacks during the Island episodes).
Heroic Sacrifice: Played for laughs when Nemo volunteers to 'bite the bullet' and try Grandis' cooking first before any of his other officers. To the shock of everybody, he genuinely finds it delicious.
Last-Second Word Swap: When pursuing King, Grandis tells Nadia to "hand over the Blue Water... I mean, give King to me!"
Lethal Chef: Played straight with Nadia (and justified by Nemo to Electra because of her circus upbringing), but oh so gleefully subverted with Grandis.
Meat Versus Veggies: In the kitchen, Nadia objects to the preparation of meat or fish, telling Grandis they're cadavers.
Mass "Oh, Crap!": The entire Nautilus senior staff's reaction to Grandis' first dinner aboard (see the screenshot). Hysterically, Grandis doesn't even register any of this, because all her attention's on Nemo.
Not So Stoic: Even Nemo looks like he wants to take his chances outside in the briny deep after getting one look at Grandis' cooking.
Non-Uniform Uniform: Complaining her new work uniform is too restricting, Nadia cuts the legs and the sleeves of her trousers and the belly of her vest, resulting in her having mini-shorts and a midriff-vest, justifying herself by stating the standard uniform.
Not Used to Freedom: Nadia choose to work on the ship since she can't stand the thought to not work:
Nadia: I am accustomed to working for my food and shelter, and I don't plan to stop that now!
Only Sane Man: Marie, the 4 year old, is literally and ironically the only sane and mature one during the dinner disaster and Grandis hunting King.
Punishment Detail: The pursuit of King and the chaos that ensues earns Nemo's stern disapproval...and lands Grandis and the Gang on floor scrubbing duty.
Ragtag Band of Misfits: The Grandis Gang and the kids in comparison to the Nautilus crew. Best summed up by the ship's Navigator when he sees Grandis giving chase to murder King (and everybody running after her):
Navigator: Who are these people?!
Speech-Impaired Animal: In the class Electra is giving to Marie, King is shown outsmarting her, a fact used by Electra to push Marie to work more.
Supreme Chef: In one of the biggest shocks of the entire series, Grandis of all people turns out to be one.
The World Is Not Ready: Nemo's discussion with Jean in Engineering reveals this is his motivation for keeping the fight against Gargoyle a Secret War instead of seeking out human governments and allies. Nemo knows it's inevitable that humanity will eventually develop technology on par with the ancient Atlanteans within the coming centuries. But he does not want 19th Century humanity gaining access to it and fears such technology would only be used for death and destruction. Considering the show takes place during the height of the New Imperialism period (eventually leading th the First World War), his concerns aren't unwarranted (and it's also being influenced by seeing first hand what Gargoyle has done with the Atlantean cultural legacy).
Wrong Genre Savvy: Jean still thinks he's in a Steampunk action-adventure story and doesn't realize that Science Fiction has entered the narrative. However, this is a unique justification of this trope (and this repeats all throughout the show). At the time Nadia takes place, both real-life nuclear physics and the science fiction genre are both in their early, formative stages. So, Jean can't recognize the clues and Tropes the audience does because in-story he lacks the context modern viewers take for granted.
