What exercises can I do to strengthen the joints, particularly the knee joints?

Full range of motion is ideal for keeping the joint healthy. The knee joint is best described as a  hinge joint, not a ball and socket joint. It bends and straightens. -with a slight rotational torque. I see many people warming up by circling the knees, a movement that is  made possible by the ankle/foot complex and does little for the knee joint. It would be better to flex and the knee and extend the knee - being careful not to hyper-extend -  slowly increasing weight and speed commensurate with one's fitness level and the overall requirements of one's practice or sport.