Learn English – the meaning of the quote “A chain is only as strong as its weakest link

from my perspective (American dialect):

To "get going" means to take concrete actions to prepare to leave. It can also mean "to hurry up." So, for instance, if I have a lot to do today, and I am dawdling over my breakfast and drinking my second cup of coffee, someone might say "you should get going!" But they probably wouldn't say "you should go!"

So "I should get going" is something I'm more likely to use when I'm implying that I have somewhere to go and stuff to do. "I should go" I'm more likely to use when I should leave. (For instance: we are at your house, we just had an unpleasant conversation, and I feel unwelcome: "I should go." I am at your house for a dinner party and it's time for me to go to bed "I should get going.")

Summary: "I should get going" implies I "should" because I have somewhere else to be, whereas "I should go" implies I "should" because it's important that I not be here. This is subtle but becomes more strong when we change "I" to "you."

As far as "I should be going," it's something I hear in movies but I don't personally say; maybe it is more British, or maybe it is just old-fashioned.

People here often grumble when a questioner hasn't done enough work on their own before asking us to help. But in your case I think you've done too much!!

Bronte is doing something unusual here. Imagine you've been looking after a neighbour's child all morning and he has been driving you mad. When telling a friend about it later you might imitate the child's whine and say, "Why haven't you got any Coke? Why can't I go to Dweezil's house? When will Mum (Mom) be home?" Or you could put on exactly the same voice and say, "He wanted some Coke. He wanted to go to Dweezil's. When would his mum be home?" Have you heard that being done? It is a bit illogical: the child didn't actually say the words "He wanted" or "his mom". Nonetheless, I've certainly heard myself doing it.

So. The aunt didn't actually say "She regretted..." Bronte uses the speech marks (inverted commas) in such an unorthodox way because she wants the reader to imagine her aunt's voice.

What her aunt actually says is something like this: (You're probably way ahead of me and don't need the rest, but I'll do it anyway!)

"I regret being under the necessity of keeping you at a distance; but until I hear from Bessie, and can discover by my own observation, that your are endeavouring in good earnest to acquire a more sociable and childlike disposition, a more attractive and sprightly manner-- something lighter, franker, more natural, as it were--I really must exclude you from privileges intended only for contented, happy, little children."

A couple of things you queried:

to be under the necessity of: needing to or having to

endeavouring in good earnest: trying seriously

from privileges intended only for contented, happy, little children: not "from things that only kids enjoy" but from things only children who act like children - proper children, in her view - deserve.

The aunt sounds pretty loathsome. As you say, she and Jane clearly dislike each other. I should read it.