We study the nonlinear stochastic heat equation in the spatial domain R, driven by space-time white noise. A central special case is the parabolic Anderson model. The initial condition is taken to be a measure on R, such as the Dirac delta function, but this measure may also have noncompact support and even be nontempered (e.g., with exponentially growing tails). Existence and uniqueness of a random field solution is proved without appealing to Gronwall's lemma, by keeping tight control over moments in the Picard iteration scheme. Upper bounds on all pth moments (p >= 2) are obtained as well as a lower bound on second moments. These bounds become equalities for the parabolic Anderson model when p = 2. We determine the growth indices introduced by Conus and Khoshnevisan [Probab. Theory Related Fields 152 (2012) 681-701].