Okay, so apart from the basics of getting the swimsuit and lining cut out, there was a small sample to be made.
On the neckline of the bodice, the elastic is applied differently than on the other edges. The front neck edge does not have a visible zig or cover stitch holding the elastic in place. The lining is used to bag out that edge. It is the left edge in this photo, where you see the zig on the blue lining.
You can see here, that the neck edge is bagged out but the armhole edge has a visible zig stitch
I made a sample before starting.
For the sample you need a piece of the fashion fabric, a piece of lining and the elastic. I thread marked the line in yellow so you can see it.
The lining and the fabric are placed right sides together. The elastic will be serged to the two fabrics, keeping the one edge of the elastic right against the line that is marked in yellow.
You could zig it on if you do not have a serger.
The final step is to control the edge by understitching. I am going to zig the elastic to the lining and in doing so, that will keep the lining and elastic in place on the inside of the garment.
The lining and elastic seam is under the foot and the fashion fabric is to the right of the foot while you are stitching.
This is when you can test your machine for the best zig width and stitch length, as well as to see that you need a new needle! (Boo, skipped stitches are not good!) That is why a sample is a good thing to do! Identify issues before you work on the final garment.
Remember, the face of the fashion fabric is not caught in the zig, and when it is all laid in place, you get this nice clean edge.
This, by the way, is the same technique I use on a tailored trouser waistband. Instead of bagging out the top edge of the waistband, I do a variation on this, using the waistband canvas in the place of the elastic and it offsets the seam between the wool and the silesia.
I will have to look for a photo of the technique used on trousers, I am sure I have one somewhere.