If you don't mind eating outside the park, give Earl of Sandwich in Downtown Disney a try. They have a variety of hot sandwiches, which are all reasonably priced. Also, the corn dogs from the Red Wagon on Main Street are delicious. If you don't want the chips that come with it, tell the cashier and they will give you a dollar off.
I thought lunch at Jolly Holliday bakery was surprisingly good and reasonably priced. The sandwiches were big, had decent flavors going on and the turkey wasn't even the usually lunchmeat stuff but actual turkey breast. For $9 it's basically the same price as Panera, which makes it a bargain for park food. If you want to save money, though, hit Subway before you go to the park and bring it in so you have an inexpensive lunch and then you can indulge in a couple snacks (churros, ice cream, Dole Whips) during the day.
Or do what my nephew does and just eat nothing but ice cream sandwiches all day.
I actually don't find the park to be that expensive to eat at. If you are already there just grab a bite. The only cheaper options around are fast food otherwise you are looking at sit down places outside the parks which are going to cost more anyways.
If you haven't been to the parks before, then you need the quintessentials: turkey legs from a food cart, churros from a food cart, corn dogs from the Red Wagon or the Stagedoor Cafe (or Corn Dog Castle in DCA), Dole Whips from Adventureland, Monte Cristos/beignets from New Orleans Square, something from the Plaza Inn, and something from Carthay Circle Theatre in DCA.
All of this is assuming you have a healthy budget. If any of it doesn't sound appealing to you, don't get it or you probably will feel that you've wasted money.
Feel free to avoid the popcorn (despite the great smell), Fantasyland restaurants, Pizza Port, Bengal Barbecue, the Golden Horseshoe unless you're there to see a show as well, and any eatery at DCA's Pacific Wharf.
I've been pleased with the Jolly Holiday bakery, Plaza Inn, and the French Market in Disneyland. In DCA, I like the food court thing at the very back- with pizza, pasta, and the Mediterranean style plates as well as the Trattoria in Wine Country. I found the Carthay overrated as far as food goes; far too fussy for my tastes. It's gourmet, but apparently my palate is not.
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We always find that the Plaza Inn has the best deal for a filling meal. The chicken plate is REALLY good.
Also, CDA, the Chinese food on the pier is tasty and filling.
Also, for a really filling meal, the big thunder BBQ, is all you can eat chicken and ribs.
Of course Dole Whips are great, but if you're budgeting, skippable, same with churros and other minor snacks.
I haven't gone off property to eat, but I'm sure there may be options.
For me my must have food item of choice is a dole whip float. I love pineapple and the only place to get dole whip is Disneyland and Hawaii.
I like to pack my own drinks, so that they are resealable and I don't have to go hunting when I get thirsty. (Not having to pay so much is a plus too.)
Plaza Inn is really good for a filling meal and there is lots of seating.
To save money our plan is usually to eat breakfast at the hotel. Then go off property for lunch if we need a break from the crowds/walking. Take a break at the hotel (nap/pool). Head back to the park and just eat dinner there.
One option that is a bit hidden is the Tangoara Terrace over at the Disneyland hotel. I am in love with their flatbread pizzas and the size makes it a nice option to split if you are going for lunch. It's also a good reason to ride the monorail as you can hop on it and then it's just a short walk up to the terrace.
I also recommend the corn dogs as they are quite filling for the price. Also somewhere here at Micechat (sorry can't remember the name) had a great recommendation for DCA. Grab a turkey leg and a loaf of sourdough from the boudin bakery and some condiments and make yourself sandwiches. I think he said they were able to get a good 2 sandwiches plus leftovers with this option.
If you do eat in the parks, go for the soups in the bread bowls. Very filling and pretty tasty too. Available at DCA and Disneyland : )
My favorite place used to be Hungry Bear in Critter Country but not since they replaced regular fries with Sweet Potato Fries. Although they will switch out Sweet Potato Fries for Onion Rings but I'm not fond of those either.
So I eat outside the park. I actually like Tilly's, just across from Paradise Pier Hotel.