So I'm deciding to study differential topology but I'm at an impasse since I hear that it might be wiser to use Loring Tu's book on Manifolds over Victor Guillemin's book on Differential Topology.
My question is whether Guillemin's book is sufficient to see differential topology for the first time or just use Loring Tu's book on manifolds.
My Current Background: Topology (Up to continuity), Real Analysis (Only single variable thus far), Abstract Algebra (up to ring theory).
If you have working knowledge of differential geometry, or know about the derivative as a linear map and the best linear approximation, then you can start with Guillemin. I personally loved Tu; it was a joy to read and filled in a lot of holes for me. But again, I was uncomfortable with the derivative as a linear map or multivariable calculus, so I benefited from a slower start.
Also, Tu does not go over transversality; however there are exercises in which you prove it he helps you prove the transversality theorem. The other omitted topics are genericity, Sard's, and Whitney's theorems, but for the depth he goes into with all the other topics, I believe there is no better replacement. One of the best things about this book is that it covers multilinear algebra. You should also give Darling's text a look. I hope this helped.