In this paper, transmission over a single-input, single-output, time-varying Rayleigh flat fading channel is considered. Time variations of the channel are modeled by a GaussMarkov process. It is assumed that neither the transmitter nor the receiver has prior channel state knowledge. The transmitter employs pilot-assisted transmission and periodically sends training symbols for the receiver to estimate the channel. Information rates achieved by binary phase-shift keying modulation are maximized by finding the optimal pilot signal transmission period and optimal training power allocation. Both fast and slow fading conditions are studied. A significant improvement in achievable rates has been shown when training power adaptation is performed. If the power of both training and data symbols are adapted, it is shown that achievable rates can further be increased especially at low SNR values.