This paper generalizes the optimized gradient method (OGM) [Y. Drori and M. Teboulle, Math. Program., 145 (2014), pp. 451-482], [D. Kim and J. A. Fessler, Math. Program., 159 (2016), pp. 81-107], [D. Kim and J. A. Fessler, J. Optim. Theory Appl., 172 (2017), pp. 187205] that achieves the optimal worst-case cost function bound of first-order methods for smooth convex minimization [Y. Drori, J. Complexity, 39 (2017), pp. 1-16]. Specifically, this paper studies a generalized formulation of OGM and analyzes its worst-case rates in terms of both the function value and the norm of the function gradient. This paper also develops a new algorithm called OGM-OG that is in the generalized family of OGM and that has the best known analytical worst-case bound with rate O(1/N-1.5) on the decrease of the gradient norm among fixed-step first-order methods. This paper also proves that Nesterov's fast gradient method [Y. Nesterov, Dokl. Akad. Nauk. USSR, 269 (1983), pp. 543-547], [Y. Nesterov, Math. Program., 103 (2005), pp. 127-152] has an O(1/N-1.5) worst-case gradient norm rate but with constant larger than OGM-OG. The proof is based on the worst-case analysis called the Performance Estimation Problem in [Y. Drori and M. Teboulle, Math. Program., 145 (2014), pp. 451-482].