I don't know the ages of the traveling party, but I can tell you my experience.
When my kids were younger, I brought a regular school type backpack with us. I would have a change of clothes for them each in a gallon ziploc bag, plus sometimes a clean tee shirt for myself (never know when you're going to spill ice cream or something!). Also, small items for while we may be sitting in a restaurant, my son's medicines, camera, snacks, etc... As the kids got older, they would be responsible for their own small backpacks.
Now I carry a little baggalini backpack - it's cuter than a school type, smaller, but I only need my stuff plus son's meds (he has food allergies). I think I got it from ebags.
I have had no problems on most rides and shows at all! Even on space mountain or something, you can put the bag down by your feet. It's no problem at all. Also, if it was just clothes and stuff (not my wallet or anything valuable) I just would leave the backpack hanging on the back of the stroller when we would go on a ride or to a show. Never have I had a problem with anyone taking it - although once my son left a big lollipop in the stroller basket that he had been licking, and when we came back THAT was gone - eeeew. Some little kid probably took it without the parents knowing (it was down low). Yuck yuck yuck.
I was just wondering what everyone else uses to carry their necessities around with, i.e. backpack/purse or fanny pack. I think we are going to use a backpack, but where do we put it when we go on the rides? and what about the heat/humidity with something this large?
