Be sure you are using a ball-point needle. Try setting the upper tension looser. You also may need to increase the presser foot pressure on your machine (consult your owner's manual if you don't know how to do this). Using a closed-toe presser foot (vs an open-toe embroidery foot) may help control the fabric and prevent it from puckering under the needle. Also, reducing the width of the zig-zag may be helpful.
Some sewing machines have an overlock presser foot. Effectively, there is a thin piece of metal in the center of the presser foot. The needle will straddle that piece of metal in a zig-zag so the upper thread has a little bit of slack in each stitch that will let the fabric lie flat. It is a very nice tool to have especially when zig-zag finishing raw edges too.
C9store overlock presser foot how to video
Some sewists will increase the fabric's "body" by inserting a layer of wash-away or tear-away stabilizer. If it is only a small project, a piece of paper added to the seam (like adding machine paper) will help stabilize the fabric.
One last thing to try would be loosening the bobbin tension. However, there is a reason it is the last effort, because most sewists are uncertain about messing with the bobbin and it is also requires "minute" adjustments to effect a big change.
How can properly adjust my tension
Understanding Bobbin Tension