I want to start studying differential geometry but I can't seem to find a proper starting path. Whenever I try to search for differential geometry books/articles I get a huge list. I know that it is a broad topic, but I want some advice for you regarding the books and articles. I want to learn differential geometry and especially manifolds. I have some background in abstract algebra, linear algebra, topology, real/complex analysis.
I would recommend Lee's book "Introduction to Smooth Manifolds." It's a long book but is comprehensive, has complete proofs, and has lots of exercises.
M. Spivak, A comprehensive introduction to differential geometry, Publish or Perish, Wilmington, DL, 1979 is a very nice, readable book.
If you prefer something shorter, there are two books of M. Do Carmo, 1. Differential geometry of curves and surfaces, and 2. Riemannian geometry.
I recommend an older book, Notes on Differential Geometry by Noel Hicks. What I like about it is that it starts with manifolds embedded in $R^n$, and shows how all of the concepts of differential geometry naturally arise there.
Sternberg's Lectures on Differential Geometry (AMS Chelsea) are wonderful and treat more than "just" Riemannian geometry.
I'm studying differential geometry in order to learn general relativity. My advisor recommended the following book: Semi-Riemannian Geometry, With Applications to Relativity, by Barrett O'Neill.
Let's suppose you can either read Russian or French, I would recommend M.Postnikov's Lectures on Geometry 3 and 4, this is really the most coherent book I've read. Okay, it's a series,though...
I would go with John Lee's Introduction to smooth manifolds and back it up with doCarmo's Riemannian geometry. If you wish to delve further into Riemannian stuff, go for the classic Comparison theorems in Riemannian geometry by Cheeger and Ebin.