I LOVE thrift store shopping and these little shorts are exactly the reason why. The shorts were originally a boys 18 shorts, high quality, usually very expensive but I got them for around 25 cents.
I have a pair of shorts pattern that I made using craft paper and a pattern I had in my stock pile. It's the size I need it and ready to use. I cut the seam in the shorts inside the crotch of the pants and laid them out side seams together. My pattern is laid with the side seams together as well. I cut away.
I did end up cutting a small piece of an existing pocket hole, so I stitched it up and cut away any excess pocket fabric on the inside.
I do french seams because they are easy and durable. Open each side of the shorts up, lay them wrong sides together and sew the outside edges close to the edge on both sides of the short.
Press out your seams right side together and sew 1/8" further in than the previous stitching underneath. The result is a seam that is not exposed and will never fray.
Measure your elastic this way. Have around 1" that over laps so that the elastic can be connected securely.
Attach the ends of the elastic by over lapping them around 1" and sewing multiple stitch lines to connect the two.
Place the connected elastic in the back of the pants on the inside right at the edge. You will then attach the opposite side of the elastic to the front of the pants near the edge. Attach by pinning. You are basically pinning all 4 "corners" together as shown above.
Sew the elastic at the edge of the shorts about 1/8" from the edge, stretching the elastic as you go from "corner to corner". You will then do an overcast stitch around the edge to prevent fraying.
Fold the elastic down towards the inside and sew along the edge, stretching as you go.