When I went for the first time without my family (so it was me + one million of my marching band friends), I knew I'd be running around everywhere and switching groups a whole lot. So I actually sewed up a little bag shortly before leaving, and I still occasionally use it. It had a zipper, was about 6-8 inches long, 4 inches high, and had a very short strap. I wanted to wear it just under my arm so I would always be in charge of it. It had just enough room for the essentials- wallet, phone, camera, any 'lady things' I may need. It could actually hold quite a lot.
ANYWAYS, I found it was a perfect size, and most (if not all) rides have SOMEWHERE you can put a bag, usually a little mesh pocket in front of or next to you. But any ride that doesn't have one, you're usually safe putting your leg through the strap and zipping it up. I was nervous about that at first, but my dad has always either worn sunglasses on a ride or somehow held onto them, and he has never ever lost a pair (on a ride :zipit
. If I recall, Splash Mountain and other water rides don't really have an area for bags, but if your stuff is protected (ziplock baggies?), just shove it between you and your partner, or you and the side, and it'll be fine. I've never lost a bag, and I'm a hardcore rider.