  In this paper we develop a new approach to nonlinear stochastic partial
differential equations with Gaussian noise. Our aim is to provide an abstract
framework which is applicable to a large class of SPDEs and includes many
important cases of nonlinear parabolic problems which are of quasi- or
semilinear type. This first part is on local existence and well-posedness. A
second part in preparation is on blow-up criteria and regularization. Our
theory is formulated in an $L^p$-setting, and because of this we can deal with
nonlinearities in a very efficient way. Applications to several concrete
problems and their quasilinear variants are given. This includes Burger's
equation, the Allen-Cahn equation, the Cahn-Hilliard equation,
reaction-diffusion equations, and the porous media equation. The interplay of
the nonlinearities and the critical spaces of initial data leads to new results
and insights for these SPDEs. The proofs are based on recent developments in
maximal regularity theory for the linearized problem for deterministic and
stochastic evolution equations. In particular, our theory can be seen as a
stochastic version of the theory of critical spaces due to
Pr\"uss-Simonett-Wilke (2018). Sharp weighted time-regularity allow us to deal
with rough initial values and obtain instantaneous regularization results. The
abstract well-posedness results are obtained by a combination of several
sophisticated splitting and truncation arguments.
