In this paper, we study the optimization of the achievable rates of frequency-division-duplex (FDD) massive multiple-input-multiple-output (MIMO) systems with spatially correlated channels, by designing the downlink channel training sequences and the uplink channel feedback codebooks. In particular, the optimal channel training sequences and a Karhunen-Loeve transform followed by entropy coded scalar quantization codebook are proposed to optimize the achievable rates. We compare our achievable rates with time-division-duplex (TDD) massive MIMO systems, i.i.d. FDD systems, and the joint spatial division and multiplexing (JSDM) scheme. It is shown that, the rate-gap between FDD systems and TDD systems is significantly narrowed. Compared to the JSDM scheme, our proposal achieves dimensionality-reduction channel estimation without channel pre-projection, and higher throughput in general, though at higher computational complexity.