Is there possibly a shop where I can sell materials and food for a higher price than regular merchants like Beetle? Or do they all offer the same prices?
There is no shop per se, but there are people that will buy your things for more money. This is not an exhaustive list, but any means, but some examples are...
Fireproof Lizards can be sold to a guy in the mines to the south of Goron City. He'll first give you an item, and then will buy three lizards at a double-cost rate to what you sell them for to merchants.
There's a guy in one of the stables that will buy your Gourmet Meat at 100 a pop, though you can only sell to him a few times a day.
A Gerudo woman in Goron City will buy a random type of gem in batches of 10 for 200 rupees more than their normal sale price. However, you don't know which gems she'll ask for.
I know of two other NPCs that will trade items for rupees.
- In Rito Village, there is a lady who will buy baked apples from you. The more baked apples you give her, the more rupees you get per apple. Apples are fairly common, and you can bake up to 10 at a time if you have access to Death Mountain.
- In the stable just before the entrance to the Gerudo Desert, there is a man who will buy 55 Rushrooms from you for 330 rupees (by comparison, most merchants pay 165 for the same amount).
I don't think this one has been mentioned yet, and correct me if it doesn't really fit the OP's questions, but after you complete a certain quest in Zora's Domain, you can trade 10 Luminous Stones to a guy for 1 diamond which sells for 500. Luminous Stones only sell for 30 each so that's a pretty good deal. And it's something that can be repeated forever, as long as you can find a good supply of stones.