I don't know how much I can spoil something that first aired in 1982, but for the sake of those who don't like long posts, I will .
Most annoying thing about this episode? The TARDIS translates all languages. So why can't everyone understand the Australian guy? It just looked like an opportunity for the writer or whomever to say, "Look! Tegan's Australian! She speaks an Aboriginal language!" Which, since this guy was from far in the past, would likely have been so different that one of his own modern-day people might not have understood him. Even supposing his dialect had survived. Okay, I know I'm not into this show for the science....
This was Peter Davison's second story, with companions Adric, Tegan, and Nyssa, but incidentally was the first Davison story to be filmed. The main thought that cropped up for me while re-watching is that Tegan serves a similar purpose to Rose in the new series; she is the human point of view, the person I most empathized with. She's the only Earth human in the crew; Adric came from E-Space and Nyssa from Traken, and besides that both of them have outsider personalities. Tegan is more "ordinary"; she was a air hostess, ending up with Doctor Four pretty much accidentally, and at the beginning of this episode, Five is trying to take her home, back to Heathrow Airport. She doesn't really want to adventure across time and space.
If I'd seen this episode without seeing others, I would really dislike Adric. (He's never been one of my favorite companions, and it's nice for me to figure out why.) He's easily sucked in when the alien Monarch appeals to his vanity about his intelligence, to the point of trying to make decisions for the rest of the crew. It's good that he has his own mind, and doesn't blindly follow the Doctor, but I still found his actions off-putting, more so than when Turlough, later, practices all kinds of betrayals; because Turlough is doing it on purpose for a reason that makes sense to me. Both Adric and Turlough were arrogant, but Turlough's arrogance seemed more forgivable to me because it arose, ultimately, out of fear, and I kept hoping he would redeem himself. Adric, I didn't care so much if he redeemed himself.
Nyssa, in this episode, shows all of her most notable traits: coolheadedness, scientific knowledge, and good judgment. She also seems aloof from the rest of the crew, which in a way makes her personality closer to the Doctor's.
Tegan panics, gets upset about things, gets frustrated...while watching her, I felt that her essential trait is honesty. She doesn't even conceal her feelings when that might make a situation easier. She doesn't doubt herself, though she will mostly accept reassurance from the Doctor, if he tells her something isn't as it appears. She manages to move the TARDIS outside of the alien spaceship, trying to warn Earth of their approaching danger. This endangered the rest of the crew, showing her loyalty is to Earth, not to them. To her eyes, none of the others seemed to care that the Earth might be doomed.
Five's coldness shows in this episode, and his scientific curiosity, and his annoyance with all these children with whom he's been saddled. His compassion we see only briefly. I recall there was more of that in the next episode, "Kinda." I shall have to see if my memory is correct.
"He knows I'm no good with my hands."
--Adric, "Four to Doomsday"