Funnily enough both the stories I got through recently were landmarks in the history of the series. The Rescue, a two-parter, was the first introduction of a new companion - followed immediately after The Dalek Invasion of Earth. I know I said I wasn't going to watch any more Hartnell, but it was only two episodes, so surely it doesn't really count. More significantly, The Power of the Daleks was the first introduction of a new Doctor, as Patrick Troughton took over from William Hartnell.
The Rescue is pretty light stuff, but fortunately survives in its entirety on video (if rather poor quality). The Tardis crew find a crashed spaceship on the planet Dido with only two survivors, apparently menaced by two peculiar monsters, neither of which turns out to be quite what it seems. One of the survivors is killed off, the other becomes the new companion, Vicki. What I liked about it most was the Doctor's knowledge of the planet as it had been on a previous visit - and then he finds that it's all changed as the natives have been (almost) wiped out. Also, of course, the travellers' adapting to Susan's absence at the beginning, and the confrontation scene between the Doctor and the main villain at the end. The monsters, I'm afraid, were a bit silly, and there were a couple of implausibilities in the plot.
I loved The Power of the Daleks, sadly available on audio only (or BBC photonovel here). I was busily spotting foreshadowings in the first couple of episodes - Lesterson, the scientist who has recovered a crashed Dalek spaceship, is a combination of Henry Van Statten (from the Ninth Doctor story, Dalek) and Davros (from Genesis of the Daleks, not the later inferior versions), and some lines seemed to me to have been lifted direct from here to the later stories.
The suspicion of Ben and Polly as to the credentials of the new man in the Tardis are entirely understandable, particularly given his habit of referring to "the Doctor" in the third person. But confusion of identity is rather a theme in the story anyway: the Doctor is immediately taken by the colonists of Vulcan to be the Examiner from Earth; the Daleks are pretending to be helping the humans (few more chilling lines than the mendacious "I am your servant!" chant which ends episode 2); the humans themselves are so factionalised that nobody seems entirely sure who is on which side.
Robert James as Lesterson was particularly good, undergoing transition from blinkered scientist, to seeing the error of his ways, to breaking down completely. I was also impressed by Pamela Ann Davey as Janley, an actual serious role for a female character. Polly does not appear in epsiode 4 (presumably Anneke Wills was taking the week off? Obviously anticipated since she is kidnapped half way through the previous episode); Ben, irritatingly, keeps wanting to go back to the Tardis and get out of the place. But Patrick Troughton's Doctor, perhaps a little uncertain at first (and hiding behind that annoying habit of playing the recorder) comes into his own pretty quickly, and by the end of the story you know who's Who.