The zipper foot is made specifically to sit close to the zipper teeth without touching them, so the machine sews straight along the tape. The key is positioning the zipper properly before you start: align the zipper teeth with the groove (the channel) you find under the foot. If you look closely, the foot has a central guide, and the teeth need to stay inside that guide. Most of the time it comes out crooked because the zipper isn't secured well at the beginning, so the fabric shifts while you're sewing.
Before you start sewing, pin everything with pins perpendicular to the zipper, not parallel. Position the foot so the needle enters the fabric about half a centimeter from the teeth (it depends on your specific Singer foot, but it's roughly there). Lower the presser foot, hand-stitch a few stitches by turning the wheel, then proceed slowly. Don't rush: the secret is low speed and stop often to check that the zipper isn't shifting.
If you really can't get the hang of the specific zipper foot, a lot of people prefer the universal foot with the adjustable side guide, which is more versatile. But honestly, once you figure out how to align the zipper foot, it becomes automatic. Try it on a scrap piece of fabric before you attach it to your new clothes.