You'll be fine. Just about every eatery has GF options, some better than others, but you won't starve. A lot of the GF menus are shown on the WDW dining page, just select a place, click menus, and scroll to the bottom. Each eatery has either a printed 'allergy menu' you ask for or, since it's your first visit, simply ask for a chef or manager and they'll come out and explain your options. An important thing to remember is that all allergy meals are cooked fresh and separately by a chef. As a result they take longer than an 'assembly line' meal prepped by food workers. At quick or counter serve that may mean that the rest of your party gets served before you do or their food sits on the tray waiting for yours. I've had times where my food takes long enough that the others are half done eating when I get my meal. This doesn't happen at table service, where the kitchen 'times' everything to be ready at once.
One note: the 'non-Disney' places (mainly at Disney Springs & World Showcase) usually don't post GF menus, but a lot of them can 'accomodate'. You have to google to see what you can find or ask at the eatery. That can make ADRs difficult in those places, but not impossible, call before you book your ADR. Finally just about any counter service that has burgers will have GF buns AND many have GF fries.
Snacks, well, there's popcorn and prepackaged 'allergy' cookies & candy (read the candy labels). But as far as fresh bakery stuff, not so much. Many things may 'read' GF, but they're dipped in common pots with cookies, etc so are not 'safe'. There are GF churros at Nomad Lounge In AK that are super delish. The Mickey ice cream bars last I heard were GF and most ice cream places can do you a 'cup' rather than a cone. And there are fresh fruit places if that's your fancy. And Starbucks. I'm not really a big snacker, though, so I don't pay close attention to the GF offerings.
I've never been glutened at WDW in 4 trips. I'm not only GF but tend to be a very picky eater and I always eat well, but it can limit my choices. If you aren't picky then you'll have more range to choose from. We also only eat one meal in the Parks as we have our trailer at Ft Wilderness.
So, places I have eaten. Most are table service since we just eat the one meal in the Parks. Check the menus, they may have changed since we ate there;
MK
Skipper Canteen
Be Our Guest
The Plaza
Pecos Bill's
EPCOT
Sunshine Seasons
Cantina de San Angel
Hacienda de San Angel
Regal Eagle
L'Artisan des Glacé
Le Cellier
DHS
50s Primetime
ABC Commissary
Hollywood Brown Derby
Oga's Cantina (just drinks)
SciFi Drive n Dine
AK
Flame Tree BBQ
Nomad Lounge (snack & drink)
Satuli Canteen
Yak n Yeti
DTD
D'lux Burger
Raglan Road
Frontera Cocina
RESORTS
Steakhouse 71 (Contemporary)
Trail's End (Ft Wilderness)
Whispering Canyon (Wilderness Lodge)
