all advice is excellent so far...here's my 2 cents about the periodic table and how I'd advise a beginner.
Find a solid piece of the blue background in your example and assuming you didn't want to or couldn't fold the edges of the periodic table, invest in a bit of fusible web to make life much easier. I suggest Heat N Bond Lite. Make sure it's the lite because it will gum up your machine if not. There are other similar brands, but I am most familiar with this one.
Iron the back of your p. table (after printing on your computer) with Heat N Bond lite. That way, you're dealing with fabrics rather than flimsy iron ons. I'd suggest making a hollowed out rectangle matching the size of you p. table plus a little bit larger. I'd make sure that you have some plain material as a border to the table so that after you iron your heat n bond to it, you can trim away some excess and seal your edge. You'll still need to sew it, but this should help to keep unraveling to a minimum.
Iron the p. table to the background fabric and sew the border around the p.table, then proceed with your directions on the pillowcase. If clear as mud, please pm me to see if I can help. Thanks
Hope you find this helpful. Best wishes with your project & Happy Sewing.