BRAT is good for both diarrhea and vomiting (banana, rice, applesauce, toast), but i wouldnt intoduce it until the vomiting's finished. Sometimes it's helpful just to give that tummy a rest for a while. On the other hand if the child is requesting something i'd say go for it, even if it's pizza, bc if they've got a craving it's gonna stay down. Probably.
Ginger tea helps with the nausea. The folk remedy to give gingerale comes from the anti-emetic properties of ginger. Clear fluids are good. Popsicles are good. I really wouldnt give soda though--a lot of chemicals.
If your child breastfeeds, it is far more appropriate to give breastmilk than pedialyte. Breastmilk is the easiest thing to digest.