DiamondRescue
Most people feed seeds as the base diet, although a few switch between seeds and pellets or feed pellets exclusively as the base diet. I feed white proso millet, canary seed, and crushed Roudybush pellets mixed with some spirulina as their base diet, but they aren't keen on the pellets.
Different species prefer different seeds. My societies will eat several kinds of seeds (favorite to least favorite): spray millet, rolled oats, white proso millet, sesame, amaranth, canary grass, white quinoa, tricolored quinoa, etc. So far I only leave the white proso millet and canary grass in at all times, though I may add the other seeds once I get another feeder. Most finches will not overeat if a basic seed mix or pellet is available at all times, but very fatty foods may need to be moderated.
Finches should have a calcium source available at all times, especially if you have females. Cuttlebone, ground baked eggshell, crushed oyster shell, and ground limestone all work. You can also leave some other mineral sources in the cage with them, although these aren't quite as essential. I provide cuttlebone, crushed oyster shell, bird clay, a mineral block and salt block that have been scored with a knife, and bird charcoal, although they usually ignore the clay, salt, charcoal, and mineral block. Charcoal is good for digestive issues, so I'd provide it in a separate container in the case that they ever want it. The non-calcium mineral blocks aren't as important, so I wouldn't bother unless you can get them cheap.
Finches usually don't like fruit, unless you have canaries. My societies like broccoli, chopped carrot, cauliflower, kale, corn, etc and I try to give them veggies once a day. Boiled crushed egg, shell and all, is also good, although you will only want to feed it 1-3 times a week if they aren't breeding or moulting (they should get it every day in these instances).
Some finches, such as waxbills, love mealworms and other live food as well, but it isn't required in their maintenance diet.
My finches are in a 4' x 1.5' x 4.5' tall flight cage. I put the food and water down low, so they use almost all of the flight. I've changed stuff since I last posted in this thread, but here's a general idea of how stuff works:
My Flight Cage
They also get some free flight in my room, but it took them a few months to figure everything out.