Bug: After the S.S. Anne is gone, go to the guard that says that the ship is gone. After facing him, keep pressing right and then quickly push the START button. Save and reset. If everything's OK, you should continue facing right. Select a Pokémon that knows SURF and use this move. Instead of surfing to the right, you'll surf ON the guard! Press down twice, and you'll enter the S.S. Anne again.
Factual error: Go to Cinnabar Island and go to the lab. There are people there who wants to trade Pokemon. Talk to one of them who wants a Raichu. Have a Raichu with you, and trade. After all the trading animation, talk to the man again. He'll say "The Raichu you traded to me went and evolved!". However, in fact, Raichu doesn't evolve into anything.
Other mistake: Being in any route or cave, and then flying or transporting to Cinnabar Island will make the Pokemon of that route and cave appear in Cinnabar's eastern coast, even though they're not water Pokemon and of course don't belong there. This can apply to Safari Zone's Pokemon also, that is, you can go to the Safari Zone, waste your steps there, fly to Cinnabar Island, surf in the coast, and you'll find the Zone's Pokemon there, which can be caught normally, and not with the methods of the Safari Zone (rock, bait).
Bug: After entering the Cycling Road from the northern gate, go all the way left. You'll see some trees beside the barrier. Then go south, until a point water appears below the trees. If you go up again, you'll see that the trees that were at north have now transformed into water. If you keep going up, and then right so the water is not seen, and then left again, the water has now transformed into trees, again.
Continuity mistake: If you come to Pallet from the south, that is, surfing, you can see that the fat man near the lab will appear out of nowhere when you enter the town.
Bug: At the beginning of the game there's a girl near your house that says she's raising Pokemon too. If you push her to the northern exit of the town, and attempt to exit it, Oak will come and will take to to his lab. Only that this time, because of the girl, you won't be able to follow him correctly and you'll trespass some buildings until finally becoming stuck in his lab's wall.
Continuity mistake: In every gym there is a statue that lists the names of all the people who have won there. The only names that appear are the player character's and their rival's, but when you go to the elite four there are all these other trainers. How did they get past the guards when they don't have any badges?
Other mistake: Go to the item list, and select any HM. Attempt to use it on a Pokemon. Before you do, the text below will say "Use TM in which Pokemon?" However, you are attempting to use a HM, not a TM, on a Pokemon.
Plot hole: Instead of going to the SS Anne, you can have a Pokemon with Cut traded to you. This makes no sense, you skip a part of the game by doing this. Also, if you advance in the game and then return to the SS Anne, Gary will appear there with weaker Pokemon than the ones he already has. Makes sense? No.
Other mistake: Marowak's ghost is supposed to be avoided only with the Silph Scope. However if you use a Poke doll in his battle it'll escape and you can proceed the game. This makes no sense whatsoever, because if you do that and advance through the game, you'll find Giovainni with strong Pokemon, but if you go later to get the Silph Scope, Giovainni will be there with weaker Pokemon. Big part of the game can be skipped with this glitch.
Other mistake: Get a Pokémon with a Special of 21, and with a move that increases its Attack (as Swords Dance, TM03). Now do the Mew glitch, that is, fly or somehow escape at the same time one trainer sees you by scrolling the screen. Fight with any trainer, then go to Route 13, 14, 15 or 16 and search for a Ditto. Let the Ditto transform into your Pokemon with Special of 21. Now, wait for it to use the move that will increase its attack. It has to use it at least three times. Return to the route where you escaped the trainer. The menu will pop-up, exit and you'll encounter a Mew, however, it will be at least in level 10, and it will know TRANSFORM. Let it transform any of your Pokémon. Then catch it. The game will say you caught a Ditto, not a Mew. The Mew will be gone forever, and instead the Pokémon will be a Ditto.
Factual error: In Cerulean City, there's a man that will trade your Poliwhirl for a Jynx. After trading, he'll say "The POLIWHIRL you traded to me went and evolved!". He may be referring that the trade evolved his Poliwhirl. While there are Pokémon that evolve via trading, Poliwhirl does not. This is a translation mistake from the Japanese Red/Green version, using the Japanese Blue text.
Bug: After you deliver Oak's parcel, go and battle Blue on Route 22. Then return to Oak, and he'll tell you about capturing Pokemon and he'll give you 5 Pokeball. This is a hidden text. However, in the middle of the text, it becomes glitched and scrambled. No wonder why they kept it hidden.