Three Elliot sisters -- not very close it would seem. Elizabeth thinks of Anne as her father does, she is "only Anne." (Chapter 1) Not too much information yet about Mary, but it is likely that Elizabeth feels toward similarly to what Sir Walter feels about his youngest daughter: she gave all the honour in her marriage to Charles Musgrove and received none, but she had grown coarse. (Also Chapter 1).
Anne seems to have some family feeling -- she wants to stay in the neighbourhood after they let Kellynch Hall, partly to be near her younger sister. Anne's motives are mixed, however, since it is clear that she does not want to go to Bath.
The only other telling detail about the sisters that I noticed was at the end of Chapter 4: Elizabeth seems to have forgotten about Anne's engagement to Captain Wentworth and Mary never knew of it. I can understand Mary not knowing -- she was at school and was only 15 at the time. But Elizabeth might have remembered the connection to the Mr Wentworth mentioned by Mr Shepherd, especially since Anne seemed to know something about Admiral Croft, if she weren't so egocentric.
With sisters like Anne's, who needs enemies.