After working through a small part of Bott and Tu's book "Differential Forms in Algebraic Topology" I've found that the topics really interest me. However, it seems that maybe I lack some background to really be able to appreciate the material. For reference, I've taken a year of analysis (standard analysis on $\mathbb{R}$ and abstract measure theory) and algebra (undergrad level) at a good uni. I've also taken grad algebraic topology (chapters 1,2 of Hatcher) and grad algebra (dummit and foote chapters 1-12 and ch 15).
My question is, how much differential topology do I need for Bott-Tu? The intro stuff on forms in $\mathbb{R}^n$ is all fine, and I've read bits and pieces of smooth manifolds books. I often find, however, that I sometimes easily get confused on stuff like orientation and forms on manifolds (especially when Bott-Tu define integration along a fiber). It also doesn't help that Bott-Tu don't have many exercises for me to practice. Am I mature enough to read Bott-Tu or do I need more background? Is Bott-Tu regarded as a good bok? Do you have any advice for reading the book? Any help would be greatly appreciated :)
You would definitely need some basic understanding of manifolds, but I don't think you will need too much. I think you will need definitions, closed submanifolds, immersions/submersions, tangent and cotangent bundles, vector fields and differential forms. Though the book defines differential forms, it chooses algebraic approach. So it would be difficult to read if you don't know what differential forms really are. But you don't need to read the whole book on manifolds. If you will need some extra-stuff, you can always look it up.
You can also refer to this book by Loring Tu and notes by Sossinsky.