In the English dub, it is said that Planet Vegeta was destroyed three years ago, but it is later changed to twenty years ago (the latter is accurate to the Japanese original.) Interestingly, even when they did the "uncut dubbed" version on Cartoon Network, in the early Raditz episodes they still used the inaccurate "three years" quote.
One of Gohan's energy attacks knocks out a number of Recoome's teeth. From that point on, the amount of Recoome's teeth in his mouth becomes inconsistent between shots. Sometimes he's shows with a whole mouthful of teeth, other times he only has a few remaining.
When Goku visits the Snake Princess, he gets her digestive juices on his clothing, which causes some damage to them. Yet when he's running on the Snake Way in the next episode, his clothes are undamaged.
King Kai, Bubbles, Elder Kai, and Gregory are all seen on Earth during the 2008 special. But they're dead and have no way of coming to Earth - there's even a line in the Buu Saga about how Elder Kai gives his life to Goku because he is unable to come to Earth.
Tien's failed suicide attack cracks Nappa's armor. When Nappa charges at the Z-Fighters (when Vegeta orders him to stop) his armor is in pristine, undamaged condition. In the scenes that follow it's broken again. (Unlike many other animation errors, this was not corrected in Dragon Ball Kai.)
The fusion of Goku and Vegeta is called Vegito, which is a cross of their Japanese Saiyan names, Vegeta and Kakarotto. Even though the dub shortened the latter's name to Kakarot, they maintained the name Vegito for the dub, which confused many fans. The American manga corrected this problem by calling this fusion Vegerot.
The original Pioneer/Saban dub got Vegeta's artificial moonlight explanation horribly wrong. Vegeta tells Goku that Bardock was a scientist and developed the technique. This is wrong; Bardock was a very strong lower-class warrior who was obviously no scientist. Note that FUNimation has long since redubbed the early episodes, and this inaccurate quote has long since been replaced with a fairly accurate description of the phenomenon.
ENGLISH DUB: During his fight with the transformed Vegeta, Goku comes to the conclusion that the transformed Vegeta is the one who destroyed the world martial arts stadium when Goku was just a boy. This conclusion is absolutely absurd since Vegeta had never before visited Earth (corrected in FUNimation's redubbed version).
Planet Vegeta was destroyed by Frieza, but when King Kai tells Goku a history of the Saiyan race, he tells Goku that the planet's Kami destroyed the planet with a meteor. The meteor story was the widely-believed story around the galaxy, but an all-knowing galactic god like King Kai should have known better (though he could have made up the story to keep Goku from going after Frieza.)
In the original Pioneer/Saban dub, when Goku dies the first time and the corpse vanishes, Piccolo says Kami had something to do with it. Krillin asks "Who's Kami?", despite the fact that Krillin had actually met Kami at the 23rd Tenkaichi Budokai (corrected in FUNimation's re-dubbed version.)
A hole is ripped in Yamcha's orange gi due to Android 20 slicing through his chest with his palm. But after Goku's friends move Goku to Master Roshi's house to recover, the hole in Yamcha's gi vanishes.
During a scene following the Raditz battle when Piccolo is talking to Roshi, Krillin, and Bulma, he is drawn with two arms despite the fact that Raditz's attack severed his arm. He didn't regenerate it until the next episode, either.
During one of the scenes right before Frieza transforms into his second form, Vegeta is drawn for one frame with the shoulder-padded armor he was wearing earlier in the saga (he switched to shoulder-less armor after the Ginyu fight.)