Doctor Who: Logopolis (1981) written by Christopher H. Bidmead and directed by Peter Grimwade
The Fourth Doctor (Tom Baker) is finally going to fix the chameleon circuit on the TARDIS so that it can change into something other than a police box. He travels with Adric (Matthew Waterhouse) to Earth to examine a real police box as part of fixing the circuit. The plan is interrupted when they run into the Master and they accidentally pick up Tegan Jovanka (Janet Fielding) who clearly looks destined to become a new companion for the Time Lord. The biggest mystery is a strange white figure that follows them on their adventures, watching from a distance.
The show is a fairly standard Doctor Who story with a few standout features. The Master is always a fun villain--he's sort of equal to the Doctor (since they are both Time Lords) but with a decidedly evil bent. His evil scheme is a little underwhelming. Tegan has the makings of a good companion--she's feisty and willing to take the initiative, even if she isn't entirely competent (though I have to be honest, even the Doctor isn't entirely competent). The final standout is the finale, where the Doctor regenerates into the Fifth Doctor, Peter Davison. Like most regenerations, the new actor just shows up at the end of the episode. He doesn't speak and nothing about him is revealed.