Plot hole: At the end of episode 1, the 'cliffhanger' ending is that the Doctor climbs over some railings, and then, by looping his umbrella on the railing, he begins to climb down. He looks down and it seems to be a very long way down. This is resolved in episode 2 by having Glitz walk around and let the Doctor step on his shoulders. Not only is it silly, but because Glitz walked around, it also makes the previous 'cliffhanger ending' totally meaningless.
Plot hole: The Nucleus of the Swarm is referred to on numerous occasions as being a virus. A virus is the simplest form of life yet recognised and is made up of bundles of DNA wrapped in a coating of proteins. They reproduce by attaching themselves to other lifeforms and releasing their genes into an unsuspecting host cell. Viruses do NOT lay eggs. Nor do they resemble prawns.(It would be more accurate to describe the Nucleus of the Swarm as a protozoan. These are simple animal lifeforms radically more advanced than viruses, around a fifth of which are parasitic although not all cause disease).
Plot hole: Sarah is transmatted down to Voga after being injected with alien poison by a Cybermat. According to the Doctor, the transmat will recognise Sarah's human molecules and separate and reject the (non-human) poison molecules. As clothing fabrics tend to made of non-human molecules, how come Sarah and Harry didn't arrive on Voga stark naked?
Plot hole: In episode four, Merak expounds on how only the Doctor and Romana can get into the TARDIS, despite the fact that he shouldn't even know what the TARDIS is. His insight continues in episode six, where he talks knowledgeably of the sixth segment of the Key to Time, despite having been told nothing about it.
Plot hole: In part 2 it's brought up in conversation several times that all metal is rare on that planet. Handy exposition for the viewers, but it's in fact a really odd thing for the characters to say. The inhabitants aren't giving a tour of their planet (certainly not deliberately at any rate), so why would they state facts that have been well known to them all of their lives? It's a bit like meeting strangers and informing them that two thirds of the Earth are covered by oceans.