Some of the books,  that I can remember, from my PhD reading list under Peter Petersen at Ucla were:  _Characteristic Classes_, by Stasheff, _Morse Theory_, Milnor, _Dimension Theory_, Hurewicz and Wallman, _The Topology of Fibre Bundles_, Steenrod.

We both agreed that Spivak's $5$ volume _Comprehensive Introduction to Differential Geometry_ was quite useful.   There was Galot, Hulin and La Fontaine's _Riemannian Geometry_.

I'll update this if I remember anything else. 

It seems to me that Milnor's _Topology from the Differentiable Viewpoint_ might have been also, and Spivak's _Calculus on Manifolds_ probably wasn't,  but I enjoyed it. 

I also had a copy of his _Riemannian Geometry_ in manuscript form, which is now available in the GTM series. 

A good place to get these was from "Book Scientific", where Spivak himself used to pick up the phone.  They had an $800$ number. 

Perhaps finally,  I don't think it's necessarily that important whether there are alot of problems,  because you should be primarily grappling with ideas and concepts, as opposed to just working problems.   He did give me also a list of unsolved problems,  which I have long since lost (maybe that's more what you were looking for! ).

He also rated the journals, and recommended reading those.  Annals of Mathematics and Journal of Differential Geometry, there were journals from Duke and Indiana university,  there was the Pacific Journal of Mathematics,  among others. Inventiones Mathematicae is also one of the best.