The easiest way in the beginning is this..
sew your elastic onto your outer layer(s) on the wrong side.. I go in 3/8 - 1 inch depending on the pattern.. from teh raw edge. A three step zig zag works well. Then you can put your other layers together with it to complete the diaper and sew a stright stitch right beside your elastic through all diaper body layers (to connect all the layers and make it much eaiser to serge). Then serge. I think this way is difficult and fussy but it was eaiser for me to do in the beginning than the next method. Now that I've learned the next method, I'll never go back though.
But, if you can master the "blind" tecnique, it will go MUCH faster and the reults are beautiful. Lay all your diaper layers togheter like it will be when the diaper is finished. Slip your elastic between the layers and tack down one end. (3/8-1 inch from the raw edges just like above) Then, when you stretch your elastic as you sew, you sort of pull it upward a little and you sew over it by feel instead of by site. A three step zig zag is good.. as wide or tiny bit wider than the elastic. This is what most wahms use. It's called "blind" because you can't see the elastic. But, once you get the hang of it, it's super easy. Just don't try it with lastin elastic until you've got a good handle on it. a knit or polybraid will be much easier to feel. Then serge. Practice on some scraps, it's easier than you'd think.
Oh.. if you go to the free mama bird pattern, she shows you a couple of ways to do the elastic if I remember correctly.