I have few questions about studying characteristic classes.
Without (at least in the immediate future) going into the obstruction theory, do I need to concider cohomology with coefficients in arbitrary ring $\Lambda$ and treat cohomology groups as the $\Lambda$-modules. Motivation of this question is that I noticed that usually there are cohomology with coeficcients in $\mathbb{Z}_2$.
It is possible (resonable) to start studying characteristic classes and K-theory parallel? I mean, both areas are new for me, but I would like to know 'applications' of characteristic classes in K-theory. Or maybe is it too hard (not resonable) at the beggining?
1) No, not in that generality. If you want to study real vector bundles you will encounter cohomology with $\mathbb{Z}/2\mathbb{Z}$ coefficients. If you study complex vector bundles you will get $\mathbb{Z}$ coefficients.
2) It is possible and reasonable to study these things parallel. I have done so (using Atiyah's and Milnor-Stasheff at the same time). I do not like MS treatment of the construction of the classes (chapter 10 maybe?) so I would look for another source for that (for example Hatcher's notes).