I just shipped some bettas this past Monday. It was my very first time shipping, but this is what I did...
The fish got 100% water change 2 days before shipping and then I didn't feed the fish for 24 hours before shipping. When it came time to package them up I filled a turkey baster with the fish's tank water and squirted that into the shipping bag. I added the fish into the bag and then a little bit of air with an air pump. I twisted the bag and tied it shut. Then I inverted the bag into another shipping bag and tied that shut. I did this with all 4 fish and sat them inside a foam lined box. The fish took up the whole box, but if there had been space I'd of added in some packing peanuts as filler. I put a layer of newspaper over the top of the fish/bags. I wrapped a heat pack with a thin layer of newspaper and taped it to the underside of the foam lid. Put the lid on and then sealed the box up. I made sure to write "Fragile" and "This End up" although I'm not really sure how much good that does

The box was shipped Priority and made it from Ks to Ny in two days and the fish were A-Ok!