Where should I stitch new badges on my son or daughter’s shirt?

Glad you asked!  Please don’t guess; you may be asked to move badges that are out of position.2015cu-as-ath

Here are the official badge placement diagrams from The Scout Association.

Badge placement diagram for Beavers
Badge placement diagram for Cub Scouts
Badge placement diagram for Scouts
Badge placement diagram for Explorer Scouts

If in doubt, ask your section leader or a Scouter.