Reference request for some topics in Differential Geometry like connections, metrics, curvature etc.

Lee's books, as mentioned in previous posts are excellent indeed. For an introduction to Riemannian Geometry I warmly recommend Riemannian Geometry by Do Carmo. For "topics in Differential Geometry" as you put it, there is a marvelous book by Peter Michor called just that, Topics in Differential Geometry. It's an excellent book that covers just what you are looking for.

Furthermore, there are two very nice books by Morita called Geometry of Differential Forms and Geometry of Characteristic Classes.

I would certainly second the suggestion of taking a look at any book by John Lee. As Bakhoda says, Riemannian Manifolds will cover metrics, connections, etc, but there is also his book Introduction to Smooth Manifolds which is, in my opinion, one of the greatest math texts ever written (alongside Aluffi's Algebra: Chapter 0). I have had my copy of Smooth Manifolds rebound because I use it so much.

Bott and Tu's book Differential Forms in Algebraic Topology is really quite brilliant, and will eventually cover the characteristic classes that you so crave. Milnor's book on Characteristic Classes is good, but if I might make a suggestion that is a little off topic, I would suggest that you read Milnor's book on Morse Theory. This is the Ender's Game of math books: Every geometer should read it, it will blow your mind, and it will change your life.