If you do decide to test sew on paper, use an old needle, or if you have to use a new needle, replace it before you sew on the nylon. Paper dulls the needle quickly (just like paper dulls scissors, which is why you never use your fabric shears on paper). As noted above, a Microtex needle is recommended for ripstop nylon, and I'd definitely recommend it since it will pierce the fabric more cleanly and combined with a longer stitch length you'll have a stronger seam.
Speaking of the needle, you also need to make sure it's mounted correctly, as not all sewing machines have you face the flat side of the needle in the same direction. In my old Singer (222K) the flat side of the needle goes to the left. In my newer machines and my vintage Bernina, the flat side goes to the back. It's not foolproof -- it's easy to put in wrong and then the stitching isn't right. Oh, and not all sewing machines take the same length of needle, although the vast majority do. I was once given a vintage Singer that took a specific non-standard needle size, unfortunately it was a long time ago and I was inexperienced with old machines and got rid of it, not realizing I could probably find the needles online somewhere.