Active reconfigurable intelligent surfaces (RISs) have recently been proposed to compensate for the severe multiplicative fading effect of conventional passive RIS-aided systems. Each reflecting element of active RISs is assisted by an amplifier such that the incident signal can be reflected and amplified instead of only being reflected as in passive RIS-aided systems. This work addresses the practical challenge that, on the one hand, in active RIS-aided systems the perfect individual CSI of the RIS-aided channels cannot be acquired due to the lack of signal processing power at the active RISs, but, on the other hand, this CSI is required to calculate the expected system data rate and RIS transmit power needed for transceiver design. To address this issue, we first derive closed-form expressions for the average achievable rate and the average RIS transmit power based on partial CSI of the RIS-aided channels. Then, we formulate an average achievable rate maximization problem for jointly optimizing the active beamforming at both the base station (BS) and the RIS. This problem is then tackled using the majorization--minimization (MM) algorithm framework, and, for each iteration, semi-closed-form solutions for the BS and RIS beamforming are derived based on the Karush-Kuhn-Tucker (KKT) conditions. To ensure the quality of service (QoS) of each user, we further formulate a rate outage constrained beamforming problem, which is solved using the Bernstein-Type inequality (BTI) and semidefinite relaxation (SDR) techniques. Numerical results show that the proposed algorithms can efficiently overcome the challenges imposed by imperfect CSI in active RIS-aided wireless systems.