If there is a very low chance of rain, I take my Dooney HM Pouchette crossbody bag. It's just the right size for a wallet, phone, some individual hand wipes, and any other small incidentals I might want for the day. If there is a good chance of rain, I take my Overland Equipment Donner crossbody bag. It's big enough to put my nylon poncho in (I put the poncho in a Ziploc bag first) and the spaces on either side that are intended to carry bottled water work great for carrying a compact umbrella. It's probably twice the size of the Dooney bag though, so I only carry it if there's a really good chance of rain (50% or higher). Unless specifically instructed (like on Soarin) to remove the bag, I just wear them for the rides.
I used to just wear cargo capris, and go bagless, but I find a small bag to be easier when navigating security at the park entrance. My BIL goes bag free, and it seems like my sister and I get through bag check faster than he gets through the no bag but metal detector line most of the time.