They're not really picky eaters so... you could probably feed them anything and they'll be fine.

[ At least his aren't complaining. ]

Most of them seem to like berries. As for your question about badges...

[ He'll turn to fumble in his pack for a second before holding up a shiny little bauble -- his Zephyr badge. ]

This is a badge. It's proof that you were able to beat a gym leader.