Revealing mistake: When the Doctor and Rose are talking at the bay, the Doctor's hair is blowing in the wind, but it's not when he is shown in the TARDIS. Considering the image of him on the beach is a projection of him in the TARDIS, the two versions should match. The actor was obviously on the beach recording his lines with Rose and back on a set doing the inside scenes.
Continuity mistake: When the two Cybermen meet the Dalek for the first time, they are exterminated. They fall directly in front of the Dalek, but when the camera angle changes to show the second Cyberman hitting the floor side on, you can see the Dalek in the background. The Cybermen both fell straight forward, so it shouldn't have been possible to see the Dalek when seeing the Cyberman side on in the next shot, as the Dalek is facing them both directly.
Continuity mistake: When Rose is hanging onto the lever and beginning to lose her grip, in close-ups of her face her body is facing the other wall of the room, perpendicular to the floor. In shots showing her and her hands on the lever, her body and head are facing the floor. (00:35:55)
Continuity mistake: After Pete tells the Doctor that Mickey went ahead because he wanted to find Rose, the Doctor asks him if he knows that Rose is his daughter from a parallel universe. The problem is that in Pete's last scene in "The Age of Steel", the Doctor saw Rose call Pete "Dad", leading to Pete departing in a hurry. (00:15:05)
Revealing mistake: In the Torchwood building, a large group of Cybermen start shooting at the Daleks and the Genesis Ark. The Daleks shoot several of them, and you can see a Cyberman that, thanks to hilarious CGI, just collapses like a sock without any physical regard that he is wearing tough metal leg armour, is floating a little off the ground, and falls way too fast than physical laws could allow. (00:25:15)