I like around the 15th of July because the Yellowstone river opens for fishing on the 15th. Stay away from the 4th and 24th of July unless you like crowds. Utah state hoilday is the 24th and this week always seems crowded. Make sure you take your kids to the swimming area on the firehole river, my kids always say its their favorite activity. Have camp reservations here is the number.
Camp Reservations 307-344-7311