My mother taught me this method of installing elastic when I was a kid and we were making hair scrunchies.
First you sew your casing channels (after turning for T&T diapers). I usually use 3/8" polybraid elastic, and sew my casings just a tad under 1/2" wide. I don't measure, but instead have it gauged off my presser foot's edge.
You can use a bodkin:
Or a safety pin:
Feed your elastic in through your turning hole in T&T diapers and your front opening in serged.
Guide it to the entry to your casings.
Feed it through your casing channel all the way to just past the end.
Then set your stitch a little smaller and sew back and forth a few times to tack down the end securely about 1/2" in from the end of your elastic (or just inset from your bodkin or safety pin.)
The secured first side:
Then detach the bodkin or safety pin from the elastic and remove it through your opening (don't leave them inside the diapers folks...). ;)
Now you're ready to stretch your elastic to your desired tension.
Stretch to where you want it and then use a safety pin to secure the elastic in-place about 1/2" from the end of your casings.
Sew back and forth again to secure your elastic in-place and remove your pin.
Now cut off your elastic leaving a little bit after the seam lines to keep it secure.
Do that for the sides and back elastic and then just close up the front opening. That is it. :)