I've just recently gone gluten free too and I found WDW to be great at handling it. When you get to the place you want to eat something, tell the person taking your order that you need gluten free. Some places will actually have one of the managers come out and take your order. Your gluten free item will be made in a separate area of the kitchen so it doesn't get cross-contaminated. One of the benefits of this process is that your food will be made fresh when you order it.
Most places have gluten free bread and buns, so you can order many of the sandwiches and burgers, just request them with GF bread/buns.
Here are a couple of good experiences I had:
Woody's Lunchbox - I ordered a GF grilled cheese sandwich from the kids menu. It was pleasantly surprised with it. I found myself wishing I would've gotten the full-sized adult grilled cheese!
Pizza window on the Boardwalk - They make a GF pizza. I was a little hesitant to order it, but I did and I'm glad I did. It was pretty tasty!
Not so good experience:
At the Roaring Fork (wilderness lodge) I ordered a GF brownie. It was a packaged brownie and was dry, dry, dry and very bland.
When looking for a "treat", I found it best to stick to things that are naturally GF. Like the krispie treats or ice cream. I actually haven't seen any "fresh baked goods" that are GF (like the fancy cupcakes or cookies). If someone else has seen these in the parks, please let me know. I would love to try them on my next trip!!!
I keep hoping Disney will step up their game in this area in the near future!