Revealing mistake: Just as the Doctor asks where the TARDIS has materialised, watch the control console wobble. However, isn't it strange that the hat-stand in the TARDIS console room remains upright thoughout...so why does the console wobble, and the hat stand not move in the slightest?
Plot hole: The Doctor should not need a book on Tibetan as he already knows the language...he went to Tibet in the Season 11 story "Planet of the Spiders." (Of course, that was Jon Pertwee, the 3rd Doctor. Perhaps losing the mastery of the Tibetan language was a result the regeneration into Tom Baker).
Plot hole: The Vogans live on a planet made of gold. They helped develop the gold weapons that defeated the Cybermen. They know the Cybermen are coming and live in a state of paranoia about that eventuality. So why in the world do they attack the Cybermen with useless projectile weapons instead of "glitter guns" or some other gold-based weapon?