Hi, Shell!
Every person has a different set of requirements for his/her day bag. I have carried everything from a diaper bag to a backpack to - now - one of the Disney Dooney & Bourke cross-body bags.
I like any bag that allows me to keep my hands free, so a sling-type bag or a backpack would be my goto styles. And yes, you can take your bag on rides with you. Thrill rides (and those coasters with inversions) provide in-vehicle storage for your items (usually a mesh or net bag that you can lean on with a leg or a tight pocket in front of you).
The beauty is that you cannot make the wrong choice. Every Disney Park offers
locker rentals; if you find that you have too much in your day bag, toss it in a locker for the remainder of your time at the Park. I've been in a situation where I actually carried a big bag --- and a smaller bag within that. When it was clear I was finished with the big bag, into the locker it went, and I happily strolled around with the smaller bag for the rest of the day.
Make sure to lay out all of your essentials and try to pare down to those. Work as small as possible, and if you can go handsfree, I would suggest doing that as well.
Have a wonderful trip!
See you later, celebrator,
-Deborah B.