FuzzyKitten .... I would suggest that your initial outline around the animals use invisible thread. I use YLI Poly quite often to quilt where I need it stitched, but don't want to see the stitching. You can stitch just outside of the satin stitch you used to applique the items down. They sell the invisible thread in two colors - clear or smoke. Sometimes the smoke color is the best so you might want to purchase a spool of each and try them out to see which you like.
Poly thread is very thin, I usually use a 60wt thread in the bobbin when I sew with Poly. Use a small eye, sharp needle - one you would use for silk. You may need to play with your tension. I strongly suggest making a small practice quilt sandwich to play around with to make sure that thread, needle and tension are all playing nice with each other.
After that you can echo quilt with whatever thread you want to use.
I love that quilt by the way!! Adorable!!