I'm making my first quilt that will hopefully look somewhat like the one below.

I am wondering about the wing coloured parts. Would I cut the fabric with holes in in to fit in the circle parts or would I just layer everything on top of each other?
I bought fusible web and tried it on a scrap piece but it doesn't stick very well and I'm worried about the pieces peeling off in the wash or something. I don't know if I used it wrong or if I got the cheap stuff. When I went to the fabric store I just got pointed to the aisle where it was and it was with a bunch of interfacing and I had no idea what I was supposed to get (as nothing had labels). I just picked up the one that looked most web-like!
Does this method work or should I just do it the old fashioned way and applique it with thread?
Thanks!!!