Book recommendation for cobordism theory

Perhaps the Notes on cobordism by Haynes Miller could be of some help too.

Another possibility (but geared primarily towards applications in symplectic geometry) is the book

V. Guillemin, V. Ginzburg, Y. Karshon, Moment maps, cobordisms, and Hamiltonian group actions. Appendix J by Maxim Braverman. Mathematical Surveys and Monographs, 98. American Mathematical Society, Providence, RI, 2002.

you ask specifically for a book; one (expensive) option is On Thom Spectra, Orientability, and Cobordism, by Rudyak, announced as

... the first guide on the subject of cobordism since Stong's influential notes of a generation ago. It concentrates on Thom spaces (spectra), orientability theory and (co)bordism theory (including (co)bordism with singularities and, in particular, Morava K-theories). These are all framed by (co)homology theories and spectra. The book is easy to use by students, for when proofs are not given, specific references are.

MathSciNet gives an enthousiastic review, "The book is indispensable for research workers in algebraic topology. The presentation of the material is very nice and thorough, and this makes the book convenient for students with preliminary knowledge of algebraic topology."

The book Differentiable Periodic Maps by Conner and Floyd is a classic reference. Despite its age, the book is very easy to understand and gives clear expositions of some important topics which are difficult to find elsewhere (such as the bordism spectral sequence and characteristic numbers of maps). As the title suggests, the latter part of the book is about the authors' research on bordism of smooth $\mathbb{Z}/p$ actions, which might be a good place to go after you cover the first two chapters.