the pump–room
A meeting place for upper-class people. Mrs. Allen's favorite place was situated closed to the main entrance of the Roman Baths. It is well-known for the power of its waters. The place is still popular today at tea time thanks to its restaurant ("Jane Austen World").
a traveling–chaise and four
A traveling chaise was a mode of quick transportation used by rich people, in the eighteenth century. This type of chaise was a closed carriage, which was equipped with four horses. The equipage, which was expensive, was generally composed of two men driving the two horses at the front and sometimes one postilion seated at the back ("Legacy Owensboro").
we shall meet them at the rooms.
The “rooms” were a place in Bath, built in 1769 by the architect John Wood the Younger. The building was divided into 4 different rooms and each were assigned a specific activity. It was the place to be for people, where they gather and exchange gossips.
"Their purpose was to be a place for dancing and music" ("National Trust").
so very much attached
"Of a person: married or involved in a romantic relationship" (OED).
to be introduced to her
"To make known in person, esp. in a formal manner, with announcement of name, title, or other identification" (OED).