From http://www.bbc.co.uk/doctorwho/classic/episodeguide:
The travellers are forced to flee in the TARDIS when they learn from the Time/Space Visualiser taken from the Moroks' museum that a group of Daleks equipped with their own time machine are on their trail with orders to exterminate them.

The chase begins on the desert planet Aridius and takes in a number of stopping-off points including the observation gallery of New York's Empire State Building, the 19th Century sailing ship Mary Celeste (the Daleks' appearance causing all the crew and passengers to jump overboard) and a spooky haunted house which, although the Doctor and his friends do not realise it, is actually a futuristic fun-fair attraction.

Eventually both time machines arrive on the jungle planet Mechanus, where the Daleks try to infiltrate and kill the Doctor's party using a robot double of him. The travellers are taken prisoner by the Mechonoids - a group of robots sent some fifty years earlier to prepare landing sites for human colonists who, in the event, never arrived - and meet Steven Taylor, a stranded astronaut who has been the Mechonoids' captive for the past two years.

The Daleks and the Mechonoids engage in a fierce battle which ultimately results in their mutual destruction, and the Doctor's party seize this opportunity to escape. The Doctor reluctantly helps Ian and Barbara to use the Daleks' time machine to return home.