Bring I know is towards the speaker, Take, they say, is away from the speaker. 'I'll bring her with me,' is that correct?
Is there some rule of thumb for us of these two words?

This is logical. The same are true in this case also: " Bring I know is towards the speaker, Take, they say, is away from the speaker" In this, "I'll bring her with me" example this is correct, if you are bringing her to here, but If you are taking her to an egreed place, wich is away from here, then you are taking her. It's also lucky to use "along" instead of "with me". So, both are correct depending on the direction. I'll bring her along. I'll take her along.