During flight they're playing with this new gizmo and use it to look at some events in history. For Vickie, the debut of the Beatles was ancient history (remember she's from the 2400's). They then land and begin exploring and of course get into trouble. But one thing they learn, through the time-space-gizmo-thing, is that the Daleks are trailing after them and intent on extermination. Except they realize, oh, the gizmo shows the past which means it's likely the Daleks are already on the planet. And, of course, that's where they are.
They have a bit of adventure with the locals on the planet, and then manage to get into the TARDIS and escape. However.. not all is so easy because the Daleks have a way to trace their travel through time and space. Thus ensues The Chase as this story is about being chased by the Daleks. They land on the observation deck of the Empire State Building.. and confuse a tourist.
They land on a sailing ship which is at sea.. there's a bit of interaction with crew members and then they leave. The Daleks arrive shortly afterward and terrorize the crew members who then abandon ship. The name of the ship? The Mary Celeste which was a historical mystery ship ...
etc, the most strange place they land is a house of horrors. It's like straight out of a horror movie. Step on the wrong board, it creaks, and vampire bats fly out of the ceiling. A guy dressed in a long coat with vampire teeth comes from out of a door and says "Good evening" in the way I used to do when I was 15 yrs old. They go upstairs and find a guy wrapped in linen with bolts sticking out his neck. etc. Very strange place and when the Daleks arrive they're somehow unable to kill any of these apparitions which itself is a strange puzzle. It leads the Doctor to believe they landed in a dimension made of the horror stories dreamed by people as if the archetypes and collective unconscious realms posited by Carl Jung actually existed somewhere. But after they leave the camera pans to a sign about so-and-so's house of horrors that had closed on such-and-such date.
They end up on a planet filled with fungus creatures and on which a huge city on stilts has been built. The Daleks land and spend awhile battling the Doctor and Companions until the robotic inhabitants of the city rescue them. It turns out the city was built by robots (Mechanoids) who had been sent by humans and that the humans never arrived to take up residence in the city. The Mechanoids intended to keep them prisoner but they had other plans. Further the Daleks invaded the city and thus ensued a battle between Mechanoids and Daleks.
And then ... an opportunity beckons. On their way to the TARDIS they come across the Dalek ship. Ian and Barbara suddenly have a longing to return home, to drink pints in a pub, eat fish and chips, etc. They plan to use the Dalek time travel machine to return home. At first the Doctor is incensed but eventually relents and shows them how to fly the ship. They make it home but it's 1965 and they'd left in 1963. This will give them a little difficulty explaining their absence and I suppose their homes might have been foreclosed on etc. But that doesn't matter to them, they're back home in London.
Which leaves Vickie and the Doctor alone. I wonder who they'll meet next?
BTW the Dalek crew is labeled as the "Seventh Incursion Squad". According to the Wikipedia entries, According to the Doctor Who Files book on the Cult Of Skaro, Dalek Sec was the Commander of the Seventh Incursion Squad and led the attack on the Mechonoids on Magella before being promoted by the Dalek Emperor. Dalek Sec was involved with the Daleks known as the Cult of Skaro who appeared in Daleks in Manhatten.