Doctor Who: The Newbies Guide
This is the fourth story in Season 20.
The Fifth Doctor is physically younger than his previous incarnations. He wears Edwardian cricketing clothes with a stick of celery on the lapel. He is less eccentric than his previous incarnations. In Earthshock he was unable to save his companion Adric from dying.
Nyssa of Traken is the daughter of Tremas of Traken. In The Keeper of Traken she helped the Doctor and Adric to defeat the Melkur, a living statue that threatened their planet. The Melkur was actually the Master's TARDIS, and at the end of that story the Master took over her father's body. In Logopolis, the Master destroyed a large portion of the universe, including Traken.
Tegan Jovanka is an Australian air stewardess who accidentally stumbled into the TARDIS in Logopolis. She has a tendency to moan. In Time-Flight she was left behind in her native time, but in Arc of Infinity she happened to run into the Doctor and rejoined him on his travels.
Vislor Turlough is an alien who was exiled to Earth. He was posing as a schoolboy at Brendon School. In Mawdryn Undead he made a deal with the Black Guardian to kill the Doctor in exchange for returning home. He did not manage to do so during that story but did stow away on the TARDIS and then ask the Doctor to allow him to join him on his travels.
Recurring and Historical Characters
The Black Guardian is a cosmic entity, being the equal and opposite to the White Guardian. In The Armageddon Factor the Doctor prevented him from getting his hands on the Key to Time. In Mawdryn Undead he made a deal with Turlough to transport him back to his home planet if he killed the Doctor.
It is well worth being familiar with Turlough's deal with the Guardian and attempts to kill the Doctor in Mawdryn Undead.