The relationship between Miss Drill and the girls. Especially in the Season 3 premiere when she comforts Mildred over Ethel's bullying. And after spending the whole episode considering leaving her job, she decides to stay - her main motivation being to act as supervisor to Mildred and her friends' new club. And in a Book Ends to the beginning of the episode, this time she joins in dancing with the girls - and looks like she's thoroughly enjoying it.
Closely related is Miss Hardbroom's decision to stay in the season 2 finale. Specifically the conversation she has with Mildred.
In contrast to Maud in the books, who doesn't make up with Mildred until after she's saved the school from disaster, Maud in the Halloween Episode actually goes straight to Mildred's room to apologise for being frosty with her earlier:
Maud: Millie? Millie, are you awake? I couldn't sleep for thinking about how horrible I was. I was tired and fed up, that's all. I'm still your friend, Millie. Nothing's changed, honestly.
When Mildred later explains the plan to save the school, Maud stops her in the middle of it to say "friends?" and they hug. Even though time is running out for them and the school (or perhaps because this is the case), Maud still wants her best friend to know how sorry she is.
In the first episode, which would also double as an Awesome Moment. Maud, who has only been at the school for one week, stands up to Miss Hardbroom and Miss Cackle - to convince them to give Mildred a chance to finish her broomstick test.
Likewise Miss Bat sticks up for Mildred too, assuring them that they're Not So Different.
The first sign of Miss Hardbroom's softer side in the first series. After Mildred has saved the school from a group of evil witches, Miss Hardbroom appears suspicious as to what Mildred was doing in the woods where she found the witches in the first place. Mildred tries to cover the fact that she was running away from school by saying she was simply going for a walk. Miss Hardbroom is still suspicious, since Mildred had all her belongings with her...but instead of the reprimand we've been expecting, she comes out with something quite different:
Miss Hardbroom: Well...whatever the reason, we were all very lucky you were there. Thank you, Mildred.