We consider a scalar-tensor model of dark energy with Gauss-Bonnet and nonminimal couplings. Exact cosmological solutions were found in the absence of potential that give equations of state of dark energy consistent with current observational constraints, but with different asymptotic behaviors depending on the couplings of the model. A detailed reconstruction procedure is given for the scalar potential and the Gauss-Bonnet coupling for any given cosmological scenario. In particular we consider conditions for the existence of a variety of cosmological solutions with accelerated expansion, including quintessence, phantom, de Sitter, and Little Rip. For the case of quintessence and phantom we have found a scalar potential of the Albrecht-Skordis type, where the potential is an exponential with a polynomial factor.