I usually starch & press my top before I layer the quilt sandwich. You really shouldn't need to press once it's layered together. It looks like there might be tiny tucks in your seam with the purple sashing. To prevent that, what I do with applique is always cut my background fabric larger than what I need for the unfinished size. Once the applique is complete, I trim down that piece to the proper size, starch & press it, then seam together with sashing/next block. I press again before layering my quilt sandwich.
The other issue is layering the sandwich. You want to tape nice crisp backing fabric to the table/floor (I use masking tape -- you could also use painter's tape). Gently lay your batting on top of that, smoothing out any wrinkles with flat hands from the center outward to the edges. Then, lay your quilt top on the pile, again smoothing from the center out using flat hands. To baste, start in the center & work toward the edges. You can use pins, thread or glue/basting spray. I've found that only stitching the edges produces bubbles/wrinkles on the back of the quilt because it doesn't allow the fabric anywhere to go. I make sure I've done a good job securing from the center outward before adding the stitches along the edge (their primary purpose is to keep the edge of the fabric from turning under & getting caught in the stitching somewhere in the middle of the quilt).
I think if you go back & starch/use sizing and press the top pretty well prior to quilting, you'll find it smooths right out. Those few tucks along the seam will become less noticeable once you quilt along that seam. Typically, you want to quilt from the center out, but because this is such a small piece, you could try quilting along the edge of the sashing & working your way down to the other edge.
Super cute elephant! Good luck with the rest of your quilt!