What Bit Do I Graduate To Next?

I have no idea what bit I should graduate my two horses to next. They're both currently using D and O ring snaffles, but I wanted to show my mare and possibly my gelding in western and barrel classes this fall. My mare is 5 now, and will be 6 before the show season, and my gelding is 11. I've looked on the the website for one of the shows I am planning on attending and it doesn't have any tack specifications. What should be the bit I graduate them to?