Watching it, it is clear from the start that it is a different sort of story - fully 10% of the story has already passed when the TARDIS crew makes its first appearance. Instead, it starts by focusing on Vicki - the new girl. The result is arguably the first modern Doctor Who story.
See, eventually Doctor Who changes to be predominantly 45 minute single-episode stories instead of the original mode of 25 minute stories of varying episode counts, and the mode for most of the series of four-episode stories with occasional 2, 3, or 6 parters, depending on what the style was. Of these, the two-parter is in some ways the most interesting - there are only 7 of them in the history of the series. (Admittedly, there are also only 7 three-parters, but stay with me) The first was The Edge of Destruction
, which was an early story that was made under such a massive pile of constraints as to be difficult to compare it to anything at all.
But what is most interesting about the two-parters is that, when credits and cliffhangers are taken into account, a two-part story comprised of 25 minute episodes comes out at about the same length as a 45-minute episode. One result of this is that this story has relatively modern pacing - certainly compared to the rest of the era, where stories are often structured with excruciating tedium.
On top of that, because the story consciously situates Vicki at its center, it is the first time since the very beginning of the series where we see the TARDIS heavily from the perspective of someone unused to it. This, keep in mind, is also the hallmark of the new series, with Rose, Martha, Donna (on her return), and to a lesser extent Amy all being introduced in that fashion.
One thing that's been very clear in the stories already watched is that Doctor Who quickly became a show with real aspirations. The Aztecs, The Sensorites, and The Dalek Invasion of Earth are all stories that, conceptually, could be made today - their basic ideas are genuinely daring. Hell, plot-wise The Sensorites basically is the plot to The Doctor's Daughter. But any of those stories would need massive replotting and reconceptualizing to actually work. The Rescue, while it would still need a good rewrite, would not necessarily need any drastic changes to its structure.
This is not entirely the consensus view of The Rescue. The major critique of the story is that its central mystery is obvious. This is hard to evaluate when you know what the central twist is, but I will point out two things in the twist's defense. First, even if you do somehow decide that Koquillion is a human in disguise and thus Bennett, this assumes that the viewer isn't supposed to get ahead of the Doctor. Which is an arbitrary decision. The reveal of Koquillion's identity hardly removes all problems the characters face on Dido. Knowing things that the characters don't is a classic technique to ratchet up suspense - indeed, the scenes of the Doctor following Bennett's trail are probably even better if you've made the leap all the way to figuring out that Bennett is the monster rather than merely in league with the monster.
Secondly, the fact that there is only one possible solution to the mystery is only a problem if the viewer realizes there's a mystery, which The Rescue avoids making obvious. This is actually a fairly savvy way to do it - instead of trying to divert suspicion from the lone suspect, divert suspicion from the mystery itself. Yes, if anyone is secretly Koquillion the evil monster guy it is Bennett, the lone character who is neither a regular nor recently announced as a new regular. But all of this presumes that the audience assumes that Koquillion is not an alien, when almost everything suggests that he is an alien.
Even after it is fairly clear that something is up with Bennett in the second episode, the assumption that he is Koquillion is in no way intuitive. The sounder assumption is that he is in league with Koquillion, because Koquillion is visibly a monster. The reveal that Koquillion is really Bennett in disguise is actually handled brilliantly, with the Doctor noting that the clothes worn by Koquillion are ceremonial garb that an alien of this planet would not normally wear. Koquillion then removes his mask and we realize that he's just Bennett in a rubber suit - the real aliens on the planet, when we see them a few minutes later, are actually completely humanoid.
This is a great reversal of expectations. Because, of course, Koquillion looks like a man in a rubber suit through the whole story. But that is in no way a clue that he's really a man in disguise. Indeed, the fact that he looks like a man in a rubber suit is, within the context of Doctor Who, extremely convincing evidence that he is not a man in a rubber suit but is, in fact, a monster. This is absolutely brilliant, and, perhaps more importantly, makes it clear that the show has known from the start that its effects were wobbly, and thus that the effects have never been about being persuasive illusions, but rather have been tools in a particular sort of storytelling. The limitations of the effects are part of the narrative structure of Doctor Who.
Which basically sums up where people go wrong with The Rescue, and to some extent with the Hartnell stories in general. They expect too little of the stories. To be fair, low expectations can often be rewarded in any period of Doctor Who. But the fact of the matter is that there's some genuinely elegant visual storytelling going on here. Consider, for instance, the early scene in which Vicki gives most of the exposition about the nature of the planet and the crashed ship. Exposition scenes are essential to Doctor Who, but brutal to stage well. This one is, frankly, the series' best yet - a long close-up of Vicki as she describes the situation that makes the scene not about the recitation of technobabble but about her desperation and fear.
If the story has one major failing it is that Ian and Barbara are a few characters too many. Their presence helps explain why Bennett is the only other human around - a short story like this suffers if it has too many characters - but the fact of the matter is, they're taking up space in this story. Ian contributes literally nothing to the plot, while Barbara gets some good scenes opposite Vicki. But looking at the story, it's clear that Barbara is paired up with Vicki not because it makes sense (Vicki stumbling upon Barbara, who is conveniently unconscious but mostly unhurt after being shoved off a cliff, is a bit of a stretch) but because someone has to give Vicki some characterization scenes. Barbara is really just filling a role that would normally be filled by another crew member if the story had room for one. All the same, Jacqueline Hill turns in a typically sublime performance.
But for the most part, this is a two-character drama of Vicki and the Doctor. Vicki is the girl who is scared and trapped on an alien planet, and the Doctor is the man who drops out of the sky and rescues her, and that's what this story is.
Interestingly, when we do cut to the TARDIS for the first time, steps are taken to increase doubt about the Doctor. He appears out of sorts, apparently because Susan is gone (there's a lovely scene where he calls for Susan to open the doors, then stops, clearly pained and embarassed at his lapse, and Barbara steps up and asks him to show her how to open the doors). Ian suggests that he's gone nuts, opting to nap inside the TARDIS instead of explore.
This is something we haven't seen of the Doctor on television since the early days of the show, and there he is more spiteful than vulnerable. Here he is wounded and clearly not firing on all cylinders. It's not until he meets Vicki and feels compelled to try to help her.
This entire dynamic is interesting, because it's also the first time the show has really asked the question of whether the TARDIS did more harm than good in arriving. At one point, Barbara mistakes a friendly alien for a monster and brutally guns down Vicki's pet sand monster. Which is an absolutely horrifying scene. But in response to it, Vicki makes the quite reasonable point that a rescue ship will be along in a few days, she's alive, she can deal with Koquillion's threats and menacing, and thus far all the Doctor and his friends have managed is murdering her pet.
This - not the Doctor confronting Koquillion later in the episode - is the story's actual climax. The Doctor is confronted with the accusation that he's doing harm, not good, and his response is, basically, to fall head over heels for the young girl, turn on the charm offensive, win her over, and save the day, all of which he accomplishes in a few minutes once he puts his mind to it. Indeed, from Ian and Barbara's perspective, he wanders off into the other room of the ship, disappears somewhere, and when he reappears a bit later he's solved everything and they're ready to go. His showing up to a concerned Ian and Barbara and basically saying "Oh, yes, I got that all taken care of" is probably the funniest scene the series has had thus far.
It is probably worth a side trip to briefly observe that this is the third story in a row to be directly connected to the UK. The last two were set there, and here the spaceship is explicitly a British ship. There's a rush of nationalism here - particularly in this story and the one before. The Daleks invade Earth, yes, but what they really want is Britain, and it's the Brits (helped by the Doctor, our honorary Englishman) who drive them off. This story is expressly set a few hundred years after the Dalek invasion, and Britain is flourishing. Given that the show just sent a major character off to rebuild Britain, it's non-trivial that it immediately goes and shows the viewers that Britain does rebuild. But, equally crucially, it does not rebuild into the Old Britain of empire. Rather, it is vibrant youth like Vicki who are the visible symbol of post-Dalek Britain. Given the parallels between Daleks and the Nazis and the fact that World War II is now firmly in the past, this message is striking, suggesting as it does a profound desire for broad social reform. And it is worth remembering specifically that the first time the Doctor really lit up and engaged with the world around him was at the prospect of the Thals rebuilding Skaro
The show, then, has quietly but explicitly allied itself with the burgeoning New Britain. Doctor Who is not only a show for kids (although it's clear that it was designed with them in mind) but for youth. Compare the ethos of a youthful, rebuilt Britain of the future with the liner notes of Beatles for Sale:
There's priceless history between these covers. When, in a generation or so, a radioactive, cigar-smoking child, picnicking on Saturn, asks you what the Beatle affair was all about, don't try to explain all about the long hair and the screams! Just play them a few tracks from this album and he'll probably understand. The kids of AD2000 will draw from the music much the same sense of well being and warmth as we do today.
The implications of this shift are completed when Vicki, our starchild, is invited by the Doctor to join them. It is very, very clear that the Doctor is a bit smitten with her (in the way that he often is with his companions). In the growing cultural divide of Britain, there is no real ambiguity as to what side the Doctor is on now. This story and the last one are the first time the show has re-invented itself, going from adventure serial to cultural icon. To a real extent, they are as important as the Tenth Planet-Power of the Daleks pair in two years, in that they firmly establish the show's capacity for reinvention and unlimited length.
And so we get, at the end of the episode, one of the series' most iconic shots - one we'll see repeated over and over again in the 46 years following this episode. Vicki walks into the TARDIS for the first time, and we see her awe at its mysteries. When Ian and Barbara fell into the TARDIS, it was a place of menace. Now it is a place of wonder, and the sequence here is the baseline that every other first-TARDIS-entrance scene is responding to.