You don't need stretchy thread. You don't need a fancy sewing machine. The one you have should be fine if it can do straight and zigzag stitches. Just sew the seam with a straight stitch first, then go over it with a zigzag.
A new ball-point needle in your machine would be ideal, but if you have to drive across town to find one, just use what you have. Practice first on something else -- if you have any even remotely similar stretchy fabric, put together some small things -- a pillow, a hat, some leg warmers -- anything to get a feel for it. If you can find a few yards of cheap (Wal-Mart near me still has a $1/yard table) stretchy material, do a mock-up of your whole suit.
Pin your seam together before you machine-sew it. Some folks even like to thread-baste (loosely sew the two pieces together by hand), but that's always seemed like too much work to me. In any case, once you know that your pieces are lined up and loosely held together, you can gently stretch the fabric as it's going through the sewing machine. Grasp it firmly both in front of and behind the needle, and support it as the machine does its work. You'll have to get a feel for how the machine feeds the fabric through; that earlier practice will have helped with that.
Sew with a straight stitch. About 10-12 stitches per inch should be plenty. If your machine has a stretch stitch, then go over the seam again with that. If not, a zigzag stitch should be fine. More stitches per inch, maybe 20, and about 1/8 of an inch wide should do. Keep the zigzag stitches to the seam-allowance side (the right side, most of the time) of the straight stitches, not right on top of them.
Once the suit is done, you might hear some of the straight stitches breaking as you put it on, but the zigzag stitches will stretch along with the fabric and hold the suit together.
Oh, and before you wear it, iron your seams flat. Makes a world of difference for a professionally finished look.
