A few years ago, I made a slingshot from stuff in my backyard and basement. Sadly, this means I cannot document the build, but I can give a quick overview on the final product.
Step 1: Handle
For an ergonomic, comfortable handle grip, I used paracord and a cobra weave. I curved the stitches as shown to fit the joints and contours of my fingers, and it is actually very soft and comfortable. To make sure the grip stayed in place, I whittled away the bark on the handle area.
Step 2: Branch
I found a fairly thick and strong Y-shaped branch off of a living tree (it was far enough out that it did not damage the tree any more than a normal pruning would). I am pretty sure it was a poplar, but like I said, this was several years ago. Any wood other than pine should be fine. Make sure it does not flex, or worse, snap.
Step 3: Thumbrest
This is a figure-8 lashing at the Y joint to rest your thumb on. I greatly helps comfort and stability to be able to steady the slingshot with your thumb.
Step 4: Elastic
I used model aircraft rubber bands (the kind you use to bungee the wing on top of the fuselage on old trainers). They were probably 15 years old when I got them, and after all of the use I put them through, they are still in perfect condition. They are much higher quality rubber than what you get on a store bought slingshot.
Step 5: Pouch
Nothing special here. Just duct tape and ingenuity.