so sorry for the slow response to your question about cycling.
With your current setup you primarily use your quads to force the pedal down starting about the 11:00 o'clock position of the pedal through the 6:00 position. The remainder of the crank ring (the cog rings to which the crank arms and pedals are attached) rotation is powered by the other leg so your starting leg is mostly at rest.
With the appropriate gear-- shoes, cleats, and cleat friendly pedals -- the rider is mechanically "locked" to the pedals. This connection allows for applying force throughout the 360 degree radius of the crank ring with each full rotation. The push down phase is a little shorter because at about 7:00 you begin to pull your foot rearward while pushing down, thus engaging your hamstring and calf muscles. This part of the stroke continues with a pulling motion that is then supplemented with a lifting motion driven by your hip flexors through the 4:00 position. Here the majority of the force comes from your hips until the 2:00 position when you add a forward pushing motion to the foot to force the foot over the top of the crank cycle. And, repeat. The other leg is doing the same thing, albeit 180 degrees out of phase.
Now compare that mental picture to the phases of a running stride, and you'll see why I suggested you can stimulate most of the same musculature when you ride.