At the very least, you're going to need a leather punch, two harness needles and a spool of sinew. You can get all that for under $20 from most online suppliers. The leather punch can either be a fancy multi-sized punch that you squeeze (like a grommet setter) or a single-hole punch that you hit with a hammer to drive through the leather. Cut out the leather like you would fabric, but leave seam allowance to allow them to overlap each other by an inch or so. It's going to fit a little differently than fabric too, so you might want to play around with holding it up against your body (especially your curves). To sew, punch a line of holes along your seamline, through both pieces of leather. Keep the holes pretty close together (.5") or less to keep the seams from rippling. Very strong hands or a few binder clips can keep the leather from moving around on you while you punch holes. Now google up 'leather harness stitch'. Basically you're going to be sewing with both needles at the same time, each one going through the same hole, in different directions. Boy, I hope this all made sense. Hope it works out great!

An alternate way of sewing is to use a sewing awl. It basically works like a stitch on a sewing machine only you do it by hand, but If you're strong enough or the leather is thin enough you do not have to pre-punch your holes (though it helps a lot if you do.)

www.speedystitcher.com/products.html

Also at Tandy they sell what they call a stitching punch (sometimes a stitching chisel) that will punch multiple holes at once. It helps keep your stitches nice and strait and neat looking.

www.tandyleatherfactory.com/products.asp

Lastly… If you want to make a vest I would definitely suggest investing in an actual pattern if this is your first project.