For sewing circles you could also try a tool that you add to your sewing machine. In some cases you can buy a brand specific tool, but you can also buy generic tools. I bought the generic one from Nancy's notions and it allows you to sew a wide variety of circle sizes. It's the only way I have ever managed to sew actual circles that look good.
For cutting circles there are several circle cutters on the market, where they use a rotary cutting blade attached to a bar, like a bar compass. Or you can also get a template that works with an 18mm rotary cutter, where you fold the fabric in half, place the fold along a marked spot, then insert the rotary cutter in the appropriate slot and cut. These templates are shaped like half circles and have grooves for a variety of sizes. When you remove your fabric and unfold it you have a perfect circle.
Rob