Finches can eat a wide range of things. I give them wheat bread, which they love! Keep doing research, but I find that if you feed your finch a variety of items (that are safe for them, they might very well live longer happier lives.
Finches eat bird seed, water, and cuttle fish. I recommend that you keep you Finch in a large bird cage so they can rome around. Sometimes you can feed your Finch a worm , but they rarely eat it.