Step 1: Pattern
You'll end up with two or three pieces: front, back, and crotch (often the crotch piece is cut in one with the back or front). Make sure the side and crotch seams are the same length on each piece so they'll match up properly when sewn.
I made two patterns, one with side seams and one without; the side seams one turned out better so all the pix, except for those on this page, are of that one.
Step 2: Cutting
Step 3: Sew Crotch & Lining
# crotch lining, right side up
# back, right side up
# crotch main piece, right side down
Using a serger, or a wide zigzag stitch, sew along the edge through all layers. Roll the back piece up in between the crotch pieces, and lay the front piece in between (probably a little rolled up too) in a comparable position between the remaining ends of the crotch pieces. Stitch these too. Turn the whole thing inside out through the sides of the crotch pieces to get your panty base.
Step 4: Leg Elastic
If you are using lingerie elastic with a clear right and wrong side, check that the decorative edge is on the right (not left) when the elastic is facing up, as this is the position it will run through the sewing machine in. If the edging is on the left, transfer the pin marks to the other end of the elastic.
Pin the end of the elastic to one end of a leg hole, and the pin you used to make the mark to the other end. The elastic should be quite a bit shorter than the panty leg; if this isn't the case try remeasuring the elastic. You want to sew the seam with the elastic stretched out the length of the panty leg, but it's hard to start that way so sew a half inch or so without stretching just to get it started. You may need to pull on the thread tails to get the elastic and fabric to start running through the machine.
Stitch with a multi-step zigzag stitch if you have it, or a regular zigzag otherwise. Once you get going, sew in segments with the elastic stretched appropriately. Pull the whole piece out to the length of the leg hole with your left hand, then grab a point 6 or 7 inches out from the needle with your right. Now you can use your left hand to guide the fabric.
At the end of the leg hole, cut the thread and the elastic. Now sew the next piece of elastic the same for the other side.
Step 5: Waist Elastic
The sewing is just like the leg holes except that you'll need to pin one additional point. The rest of the panty prevents you from holding the waist out straight, so you can't just pull the elastic to see how stretched to make it. Pin both ends, then hold them both in one hand and find the middle of the panty waist and the elastic, and pin them together there. Note that this probably won't be very near the completed side seam, as the front of the panty is rather smaller than the back.
Step 6: Finishing
The last thing is to trim any edges that stick out past the elastic, being careful not to cut the elastic of course.
Once you've made a pair or two, you'll find that they go very quickly. You can make several pairs in an afternoon, they'll be completely unique and fit like they were made for you!