Anticlimax Boss: Mewtwo. Despite the fact that the cutscene before unlocking its dungeon makes it look like the most powerful Pokémon of all, it's only Level 40 and comes after 98 floors of powerful foes like Tyranitars, Alakazams and every form of starter. You should have no problems beating it quickly, especially given the massive amount of levels you're going to gain on the way to it.
Ditto with Lugia, who comes at the end of Silver Trench, a 99 floor dungeon filled with powerful enemies tat can snip the party repeatedly and tons of traps. Add in unavoidable monster houses every few dozen floors as the cherry on top and you have a truly grueling level. Lugia meanwhile only has a weak Gust attack to deal damage, and while it does have a lot of hit points it's also weak to plenty of common types. It's telling that even after a level as draining as the above hardly stops a sufficiently leveled team steamrolling over it.
The player character and the partner. All subsequent games would end up carrying this as a tradition.
Gengar/Gardevoir. For bonus points, these two species can breed in the main series.
Fridge Horror: At one point in the story, the main character and his/her friend are driven out of Pokemon Square because of Gengar tricking everyone into thinking that the main character is the human from the Ninetales legend who abandoned Gardevoir under the impression that he/she'll destroy the world. This is pretty sad in itself, but after you complete the game and save Medicham from Wish Cave, it's revealed that Gengar is the real human from the Ninetales legend. Basically, Gengartried to make the main characters his scapegoats and make them take the fall for his crime!
Fridge Logic: Your first mission as an official rescue team involves rescuing some Magnemite who got stuck in a cave, and the reason they came to you is because Caterpie told them about you. Except Caterpie doesn't know where you live, nor even what your team name is.
Game Breaker: Due to lacking the gender mechanic that the main series games have and the tweaks this game made to Standard Status Effects, Attract's infatuation effect disables anything without the Oblivious ability for 5-6 turns. This includes bosses. Any potentially hard encounter is made trivial by hitting a boss (who tend to fight you alone) with Attract since you always go first and can shut them down before they can take you out.
Idiot Plot: The Sinister Woods arc. Gengar and Team Meanies find out that Caterpie's friend Metapod has gone missing, and challenge the player's team to a race: if Team Meanies finds Metapod first, they get the credit, Butterfree will give them a reward, and Caterpie will join their team. Butterfree is the mother of Caterpie, not Metapod, and has no reason to give out any kind of reward- and even she did, she's hardly the kind of person who'd have large amounts of cash on hand. In addition, Caterpie is a tiny kid- not the best recruit for a team planning on taking over the world.
Narm: Team Meanies is very difficult to take seriously because of their name and how up front they are about wanting to Take Over the World.
Padding: The entire chestnuts subplot. Besides being the part where you redecorate the team base, it has no importance in the overarching story and brings the game to a complete halt since it forces you to visit the same dungeon multiple times to find the Random Drop chesnuts.
Articuno is the first boss Pokémon to wield a move that strikes your whole party (Powder Snow) — be prepared to lose a few Reviver Seeds here, especially if one of your starters is Grass- or Ground-type.
Woe betide any who bring a team of a Fire/Grass to Moltres. Many have Rage Quit because of this boss thanks to most of its attacks being OHKOs to Grass types.
That One Level: Teammates cannot be brought into Purity Forest. Upon entry, the team leader will be reduced to level 1, returning to its original level after exiting the dungeon. IQ will also be reset to the initial skills until the player exits the dungeon. Any items or Poké carried prior to entry will be permanently lost. Wild Pokémon cannot be recruited in this dungeon, and facing progressively harder enemies (reaching even 30 at the last floors)