We consider the estimation of an i.i.d. (possibly non-Gaussian) vector from measurements obtained by a cascade model of a linear transform followed by a probabilistic componentwise (possibly nonlinear) measurement channel. A novel method, called adaptive generalized approximate message passing (adaptive GAMP) is presented that enables the joint learning of the statistics of the prior and measurement channel along with estimation of the unknown vector. It is shown that, for large i.i.d. Gaussian transform matrices, the asymptotic component-wise behavior of adaptive GAMP is predicted by a simple set of scalar state evolution equations and that the parameter estimates are provably asymptotically consistent under essentially arbitrary parametrizations.