Assume that $E$ is a complex oriented spectrum. Then for the corresponding generalized cohomology theory (which I shall denote again by $E$) we have the following:
$E^{*}(\mathbf{C}P^{\infty})=E^{*}({\rm pt.})[[x_E]]=\pi_{*}(E)[[x_E]].$
My problem is with the notation of the above statement. I don't understand whether these equalities refer to rings or abelian groups or modules, and what is denoted by $\pi_{*}(E)$? Moreover, why the last is equal with $E^{*}({\rm pt.})$?
These are isomorphisms of $E_*$-algebras (I wrote modules before, but it's also true as algebras), where by $E_*$ I mean $$E_* := \pi_* E \cong E^{-*}(\text{pt}).$$
To see the isomorphism in the above line, recall the definitions of the homotopy and cohomology of a spectrum. We have $$\pi_n E := [\Sigma^n \mathbb{S}, E]$$ and if $X$ is a space, then $$E^n(X) \cong [\Sigma^\infty X_+, \Sigma^n E],$$ so in the case $X$ is a point, we have $\Sigma^\infty X_+ = \Sigma^\infty S^0 = \mathbb{S}$. So $$E^{-n}(\text{pt}) \cong [\mathbb{S}, \Sigma^{-n} E] \cong [\Sigma^n \mathbb{S}, E] = \pi_n E.$$