We carry our stuff in a back pack typically. You can take your bag onto the rides. Most rides are not that extreme and you can put your belonging right at your feet. Some have pouches you can stuff your bag into but if your bag is really big it won't fit. We usually will just wrap the strap around a wrist and it's fine. If your going to be carrying a camera, it's a good idea to bring a plastic bag to put it when it rains, or on on wet rides (mainly, Kali at AKL). This works well for smaller cameras. If you have larger, more serious equipment then you might want to bring a dedicated camera bag that is weather resistant.
As far as what else we take in the park. It seems to change a little with each trip (as ds gets older), but I'd say a basic change of clothes for him, possibly even some water shoes depending on where we are going (there are splash fountains for the kids to play in). I usually bring some lite snack like crackers chips or pretzels for ds and bottled water for all of us. We still use a stroller so we don't have to put everything in the bag. Some of it goes under the stroller until we need it.
Some of the rides do. Two of the most 'extreme' rides there, IMO, Tower of Terror and Rock 'n Roller coaster do not have pockets to put your stuff in. On those, I tuck my bag under my legs and also wrap one of the straps around my wrist. I haven't had any issues with holding on to things.
There are also lockers in the front of the parks you can rent for a nominal fee. If you park hop you can take your receipt and get a new locker at the other parks. You might consider doing this if you won't have a stroller to stuff things in or don't want to lug a bunch of stuff around but want it to be fairly "handy".
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