Bellatrix is power hungry and despises muggles. Being the oldest, I think she felt that she had to live up to her family's expectations. So she followed in their path she believes in their beleifs.
I think she was drawn to Voldemort because he has the same beleifs as her, he despises muggles and muggle borns, he's power hungry and is very passionate about what he does. There are quite a lot of similarities between them, and I think it's only natrual that Bella ended up serving him.

Andromeda, well we haven't really heard much of her, but in my head she is the smartest Black sister. She's independant and does what she wants, not what everyone expects her to do. She isn't as narrow minded as her family, she doesn't think that power comes from being pureblood and believes that muggle borns can be just as talanted as the rest of them.

Narcissa grew up in her sisters shadows, she's described to have a fragile beauty and is probably then a fragile person. I think she looks up to Bellatrix and probably idolizes her, but doesn't want to risk becoming a Death Eater.
She cares more for her family than Bellatrix does, and finds that her duty to the Dark Lord, is to give him the men in her life. Lucius and Draco.