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. 

The knee is  a mediator between the foot/ankle complex and the hip. If we do not move our feet when twisting the body anywhere above the ankle, it is the knee that takes the brunt. In addition, the knee takes on the work of the ankle/foot complex which is often lacking in proper strength and flexibility and, so, not supporting the body from underneath where foundations do their best work. This lack of support plays all kinds of havoc with our practice. It interferes with grounding/stability, as well as our ability to generate power through speed and mobility. We are constantly lifting our legs from the knees and hips(or arms) instead of pushing from the feet. This leads to over-use in the hip flexors and under use in the hamstrings, a common muscle imbalance, leading to joint imbalance and, before we know it, mysterious knee pain. The other muscle imbalance issues which can affect the knee are related to the hip abductor and adductor muscles. These muscles take the leg away from and toward the mid-line of the body respectively. Imbalance here can affect the tracking of the patella or “knee cap”. 

Throw in the knowledge that the knee joint is less stable from “side to side” impact when it is flexed,  self directed impact (body weight +) when over-extended and  perpendicular impact (a strike to the knee-cap) when it is extended and you have a few more tools with which to self diagnose. 

Most importantly, stop viewing your knee as a separate entity that needs to be fixed, the proverbial squeaky wheel, instead view  the whole you…. um….dare I say it, the bike, plus the road that it rides upon.