"Seemed chiefly interested in small news about Frodo's health and doings"
This quote means that Gandalf is actually wondering if Frodo is using the ring just for the sake of his safety. This means that Gandalf cares for Frodo because a person would not just go to someone home often just to check on the person and then leave that evening or the next morning. This shows that Gandalf is looking after Frodo and has done what Bilbo has requested him to do. Also since the ring was given to Frodo, and Bilbo ask Gandalf to watch over him this is a great way of showing how he is good friend for Bilbo and a good guardian for Frodo.