Awesome Music: That intro. No matter how highly you regard the series, most fans agree that the intro and the music that goes with it was the best and most memorable thing about the show. Judge for yourself.
Jessie. She had plenty of fans, obviously, such as those who saw her as a kick-ass female character and those who saw her as hot and those who liked her romance with Jonny. Others, however, balked at the introduction of a new character to the mix, while others saw her as a Faux Action Girl (her Badass Decay in season two didn't help).
It also didn't help that in the very first episode, "Darkest Fathoms", she ends up as a Damsel in Distress, which became the impression many early reviews of the show got of the character, not giving the show a chance to show her badass moments later on in the season.
Complete Monster: The Monster of the Week from "Eclipse" was the succubus/vampire known as Elise Lenior. Initially, she appears to be a victim, an innocent young woman on the run from evil men, The twist comes when it's revealed Elise was maintaining her youth and beauty by draining the life from beautiful, innocent women and the man hunting her was trying to avenge his sister when she fell victim to her. Elise murdered the men who tried to tail her, and later mind controlled Hadji into her slave to bring Jessie to her for her food source. When her right-hand man protested about Elise trying to replace him with Hadji, Elise declared he was right and sucked the life from him as well. When her deadline of an eclipse was approaching, Elise abandoned all pretense of charm and subtlety and assumed her monstrous true form to kill everyone around and drain Jessie's life.
Ethnic Scrappy: The first season's writers actually worked to avert this trope with Hadji, only for him to fall back into it in the second season with the reintroduction of the Sim Sim Salabim elements of his character. However, he first season's success at aversion is still debatable, considering they ended up turning him into a Bollywood Nerd computer expert (although that hadn't yet become a stereotype at the time the show was airing).
Jonny and Jessie for an overwhelming majority of the fandom, although Hadji and Jessie are also well-liked, probably due to massive amounts of Ship Tease for both ships.
Apparently the series creators felt that Jessie would be the sort of girl Jonny would fall for when he got to the age were he'd be interested in dating.
Definitely sent up in an episode where Jessie and Jonny had their bodies inhabited by a pair of star-crossed ghosts. They use them to reunite and depart for the afterlife, but not before engaging in a seriously intense kiss that Hadji eventually has to clear his throat to get Jessie and Jonny's attention to break up once they're gone. After some amusing awkwardness, Jonny checks some recording equipment they had set up to investigate the haunting, only to find that only a few minutes of footage had been recorded to account for over an hour of time. Jessie teases him with a drawling, "Yeah. The earth stood still for me, too, Jonny."
For a while, it was actually possible to get Word of God, at least for the second season. The short version was that they chose to split the difference between those who shipped Jessie with Jonny and those who shipped her with Hadji.note You can tell which writer shipped which from the ship tease. The shipping wars within the fandom managed to drive some fans into taking a third option: Jessie with neither of them.
Though, to give the writers more credit, India was already headed that way in the 1990s, so showing Hadji as a computer-skilled youth was meant as a deliberately progressive portrayal.
Jonny's voice actor Quinton Flynn would go on to voice Raiden in the Metal Gear series, and mentions in this interview that he was told by the voice director (who also worked on both series) to imagine Raiden as an older Jonny Quest. Considering all the Narmandotherthings involving Raiden in that game alone (not to mention laterones), a lot of scenes from the show (plus the implications for poor Jonny) become beyond hilarious in this context.
Memetic Mutation: "(Insert action here)... so it is written IN THE BOOK OF RAGE!!!"
Nightmare Fuel: "In the Wake of the Mary Celeste" is terrifying from the start, with the titular ship and those aboard it being overtaken by an eerie circle of green light. The ending has the villain of the episode suffering the same fate.