Xanne

I just use a book bag (like a regular book bag not a book bag style diaper bag). Does your stroller have any storage underneath? 
 
Welcome to the boards!

Welcome!
My "diaper bag" was a changing pad (in a clutch style) that I purchased at Target or Walmart.  It would hold a travel pack of wipes (the hard sided ones that come with the box or wipes), rash cream and 3 or 4 diapers.  I would use the wipes to clean off tables and high chairs and carry a travel size sanitizer (they have alcohol free if you prefer for your little one).   This can easily fit under the stroller.  Outside of that, you might consider a light weight backpack to carry any additional items.  The more trips we took, the less stuff I took (even when my babies were tiny) as I discovered that all we did was haul around a bunch of stuff that we didn't use ☺
 
Have a great time and again...welcome!!

Last time at Disney my son was 14 months old. We use a very small messenger style bag (not a diaper bag) it held a change of cloths, a few diapers, snacks and the all important Binky.   
It was really easy to carry and wasn't in the way at all.  For a stroller we used our Bob stroller.  
Our every day diaper bag is a backpack style that is made by Columbia and i really like it too.
 
For me the most important thing is not to over pack for a day in the park.  Most 2 year olds cant do 8 or 10 hours so plan for 4 or 6 hours of need because at some point they are going to crash and need downtime.