There were a handful of loose ends at the end of Logopolis (such as, what happened to the Master?), and Castrovalva endeavours to address these points by picking up where Logopolis left off. And so we see the Master escape while the Doctor's companions help the newly-regenerated Doctor inside the TARDIS. Only it seems there's something wrong with Adric...
Hang on a moment though; why is Tegan going with them? Look at it from her point-of-view: she's been kidnapped (accidentally, but still) and forced to wander a white corridor labyrinth (that still upsets her, judging from her reaction in this story), then taken to an alien planet, where she demands to be returned to Earth, and then she sees the universe almost destroyed by some evil madman who ends up killing the only person she feels she can trust ("I'm staying with you," she tells the Doctor in Logopolis. "You're the only insurance policy I've got") -- except then she watches as some weird-looking plaster-of-Paris-esque dude merges with the Doctor and creates a brand new person. She's on Earth, in what appears to be her own time period; what possible reason does she have for getting back inside the TARDIS? She must have incredibly strong feelings of loyalty to someone who doesn't actually look like the Doctor. Or maybe no one's actually thought about this.
But anyway, into the TARDIS, where we start117 to see Peter Davison stretch his muscles as the Doctor -- which he does by impersonating the first three. He does a good job (although his Hartnell is the best), and it's a nice sequence that seems designed to remind the audience that there were in fact Doctors before Tom Baker. I also like the self-deprecating humor, as he first realizes in dismay that his hair isn't curly anymore, and then when he finally sees himself in a mirror. "That's the trouble with regeneration," he says despondently. "You never quite know what you're going to get." But he soon seems happy enough, as he picks up a cricket bat, and soon we see that he's going to be wearing a sort of Edwardian cricketing outfit.
The interesting thing about this story's approach to the regeneration of the Doctor is how drawn out it is. It's most like Jon Pertwee's debut, where he spent the first couple episodes of Spearhead from Space bed-ridden, but Castrovalva goes even further than that -- this is the weakest we've ever seen the Doctor, and it's explicitly because of post-regeneration trauma. This means he's not much help when the Master (via Adric, who's actually a block transfer computation projection -- just go with it) programs the TARDIS to head back to the hydrogen inrush that led to the formation of the galaxy (huh? Do they mean the universe?) -- an event that the TARDIS can't withstand. (Cue part one's cliffhanger.) This means that the Doctor is fighting not only his own healing synapses but also the Master's efforts to kill them all. They manage to escape, but the Doctor is still unwell -- and the Zero Room that he was recuperating in was jettisoned in order to escape the Master's trap.
The second half of part two involves Tegan maybe piloting the TARDIS (her abilities are deliberately questioned) to a place called Castrovalva, where the Doctor can relax. They build him a small Zero Cabinet and wheel him around the planet on a motorized wheelchair, but when the wheelchair falls in the water the motor shorts out, which apparently means the wheels don't work anymore either? So Tegan and Nyssa have to lug this cabinet around. And then the episode ends when they put the cabinet down, wander off, and then come back to find that he's not in the cabinet anymore. The fact that this cliffhanger works at all is a testament to how much effort Janet Fielding and Sarah Sutton have been putting into this thing, that this is something of a distressing moment and not just a case "oh, well OK then".
But then, these first two episodes have been surprisingly compelling. The cast is quite limited (consisting of the four regulars and the Master, with an occasional line from one other guest actor), and the focus is thus on the Doctor's regeneration, to an extent that we haven't seen before, and it's this that makes Castrovalva work so far; the series is interested in exploring its own mythology a bit, and we're willing to go along for now, to see what they come up with.
117 Well, he starts from our point of view, but Castrovalva is actually the fourth story Davison filmed (the result of several different debut story scripts falling through), which means he's had time to figure out how to play the character and thus can work backwards from that to give us what's probably a more coherent, thought-out performance than what we otherwise would have gotten.