This paper considers a single-cell massive multiple-input multiple-output (MIMO) system equipped with a base station (BS) that uses one-bit quantization and investigates the energy efficiency (EE) and spectral efficiency (SE) trade-off by simultaneously looking at the uplink and downlink transmission. To this end, we first propose a new precoding scheme and downlink power allocation strategy, which makes the uplink-downlink SINR duality hold in the one-bit MIMO systems. Then, by taking into account the effect of the imperfect channel state information (CSI), we obtain approximate closed-form expressions for the uplink achievable rate with maximum ratio combining (MRC) and zero-forcing (ZF) receivers, which, according to the duality property, can also be achieved in the downlink transmission. By employing the multiple objective optimization (MOO) framework, we focus on the optimal design for the EE and SE trade-off that jointly selects the number of active terminals, pilot training duration and operating power to maximize both EE and SE. The weighted Chebyshev method is used to obtain the Pareto boundary of EE and SE, which allows the system to know all the possible operating points and balance the EE and SE in an efficient way. In order to go beyond the Pareto boundary and actually solve the MOO problem, this problem is transformed into a single-objective problem using the a priori method such as the weighted product method, where the operating EE and SE are chosen with equal importance by adjusting the weights. Numerical results are presented to verify our analytical results and demonstrate the fundamental tradeoff between EE and SE for different parameter settings.