After cutting the elastic to length, I overlap the ends about 3/8" and loosely join them with a zigzag. This creates very little bulk at the join.
Next, quarter-mark the elastic …
… and the waist (or leg openings).
If you're new to sewing elastic while trying to stretch it at the same time, you might want to mark in 1/8 intervals instead so you have less to stretch between pins as you sew. Remember to take it easy, go slow, rearrange as necessary, and do not stretch so much that it puts stress on your needle or you will bend/break the needle. One hand behind the needle stretching from that side helps to avoid too much pressure on the needle.
Also, many times you will want more stretch in elastic in the backs of leg openings and less stretch at the front. Just remember adjust your marks to compensate for that.
Match your quartermarks from above and pin the elastic to the opening with elastic wrong side (non-plush) to fabric right side, and the picot edge away from the fabric edge.
Gently stretch the elastic between the pins as you sew (with one hand in back of the needle and the other in front).
Attach the elastic with a very narrow zigzag, close the picot edge, as shown below.
Turn the fabric over and trim anything that sticks out past the edge of the elastic. Even if nothing is sticking out past the elastic, you may want to trim to reduce bulk — your preference. When I'm sewing assembly-line style, my preference is to trim as little as possible so I only trim what's sticking out. ;-)
After trimming, fold the elastic in toward the fabric wrong side.
Gently stretch the elastic as you stitch using a very wide 3-step zigzag. Stitch at the edge of the elastic opposite to the picot edge to catch the un-picot edge of the elastic, as show below.
This is what it looks like from the right side.
The farther from the picot edge you stitch the first line of small zigzags, the more the edge of the elastic will show when you fold it inside the waist/leg, as shown below.
This is what the inside looks like when you're done (my sample is sewn 1:1 with no stretch just for clarity). Remember, if you're using bobbin thread to match the elastic instead of contrast like this, it won't show.
When you're done, the elastic may be stretched out and a bit flat, but it will snap back after a run through the laundry, or a few blasts of steam. The panties on the left below were just finished and the elastic is pretty darn flat after sewing it on. Those on the right (same size) have been through the laundry. You can see how the elastic shrunk back to size. In other words, toss what you've done into the washer/dryer first before thinking you haven't done it right. ;-)