Inspired by recent developments in full-duplex communications, we propose and study new modes of operation for cognitive radios with the goal of achieving improved primary user (PU) detection and/or secondary user (SU) throughput. Specifically, we consider an opportunistic PU/SU setting in which the SU is equipped with partial/complete self-interference suppression (SIS), enabling it to transmit and receive/sense at the same time. Following a brief sensing period, the SU can operate in either simultaneous transmit-and-sense (TS) mode or simultaneous transmit-and-receive (TR) mode. We analytically study the performance metrics for the two modes, namely the detection and false-alarm probabilities, the PU outage probability, and the SU throughput. From this analysis, we evaluate the sensing-throughput tradeoff for both modes. Our objective is to find the optimal sensing and transmission durations for the SU that maximize its throughput subject to a given outage probability. We also explore the spectrum awareness/efficiency tradeoff that arises from the two modes by determining an efficient adaptive strategy for the SU link. This strategy has a threshold structure, which depends on the PU traffic load. Our study considers both perfect and imperfect sensing as well as perfect/imperfect SIS.