There are many ways to cut circles, but if the circles you want are larger than you can cut with a hole saw, you need to make other tools either go around a centre point, or make your wood follow a centre point.
With a band saw the easiest is to clamp a piece of plywood to the saw table with a nail sticking up straight across from the cutting tip of the blade at the right radius for the circle. You need to cut straight into the piece of wood, then hook the centre point on the nail and then simply spin the wood around to cut out the circle. The key is having the flat of the saw blade exactly perpendicular to the radius of the circle, which is the same as being tangent to the curve of the circle. This will keep the blade from wandering away from the circle or cutting into the circle. The same tangent trick is necessary when trying to cut a circle with a jig saw and a 'T' guide. Keep the teeth touching the circle and the flat of the blade perpendicular to the radius and you won't have problems.
When you use a router you reverse the task, as in the photo above. Put a pin into the wood to be cut and through an extended router base. That makes the radius of the circle. If you are going to cut all the way through the wood, make sure that your pin is secured into a piece of scrap below, because as soon as you complete the circle, you loose your anchor point -- like cutting off a tree limb while sitting on it.