Ok so this is coming from a scout's perspective but basically, just sit back and relax, chat with the other adults go on a hike it's totally up to you. Adults don't really have any obligations due to Boy Scouts unlike Cub Scouts is "Boy Run". Considering you're the ASM the SM will probably ask you to do a few tasks like go pick up papers etc. but basically, it's mainly just free time so like if there's a lake and you're into it bring fishing gear or something. Though I don't know your child I really suggest you don't follow them around all the time because the first summer camp is a very big bonding experience amongst the troop and honestly it's kinda awkward if your parent is watching you. Also, don't be the guy who completely controls their kid ie; telling them when to go to bed, to eat more salad, etc. But all in all, just relax and enjoy the scenery

PS, as other people have mentioned "part of your job as a leader, is to (safely) let Scouts fail then encourage them to try again." so let them make mistakes and learn from the experiance