You know, looking back on it, I have NO IDEA what my parents did all day on I-Day. I don't even remember if we got to see them after the Oath of Office -- I THINK we did. Unfortunately, they're no longer alive, so I can't ask them.
During the day, do what YOU want to do. Don't worry about what you're "supposed" to do -- no one cares. If the picnic sounds good to you, do it. If it sounds crappy, skip it.
IF you choose to try to catch sight of your mid coming off the bus or whatever, please realize that it's meaningful to you, not your mid. He/she will be so focused on what's going on, trying to do the right thing, etc., that the very LAST thing he/she will think about is trying to see you. If it happens, you may get a smile, but that's about it. Again, I have NO memory of seeing my parents between the time they dropped me off at Halsey (in our day) and . . . not really, after the Oath.
I definitely remember many other great times, like eating an ice cream sundae to celebrate my 18th BD just before going back for Hell Night.
Be there for the drop-off and the post-Oath visit. Relax! Your child will do fine and be fine. And you'll justifiably be very proud!