Seeds
Feed a good quality finch seed mix, although this should never the the sole diet of your finches. Check that the seeds are fresh by sprouting them (put some in a plastic bag with a damp paper towel) - if they will not sprout then they are too old to feed. Millet sprays are a favorite treat of many finches but should be given sparingly, or the finches may develop a preference for millet.
Sprouted Seeds
This is an excellent way to boost nutrition as the seeds are at the peak of their nutritional value at sprouting. Ideally they should be fed just as they begin to sprout. For information on sprouting, see this page from Landofvos.com.
Greens and Fresh Foods
A variety of green should be provided, such as romaine lettuce, dandelion greens, kale, and spinach (in moderation), along with a variety of fresh vegetables and fruits (NO avocado, though). Experiment to find what they like and keep offering a variety as it may take your finches a few tries to accept new food items. Remember, though, that serving sizes of these foods should be small.
Pellets
These are excellent balanced diets and are nice to offer as part of a varied and balanced diet. Be sure to get a good quality pellet sespecially formulated for finches, and you may need to be persistent in offering pellets before your finches will try them if they are not used to eating them.
Eggfood
This is a supplement that is very important for breeding birds but that can be fed in moderation to non-breeders as well. You can by commercially prepared eggfood supplement of prepare your own (recipes linked at FinchWorld).
Cuttlebone and Grit
Some sort of calcium supplement is recommended and a cuttlebone is an easy way to add calcium to the diet, as are crushed oyster shells or even eggshells (heat thoroughly before crushing to prevent salmonella contamination). The role of grit is still highly debated. It is now quite widely accepted that parrots do not need grit, but the need for grit for finches and canaries is less clearly understood. Many avian vets now recommend no grit even for finches, as they do hull their seeds (grit is essential for doves and pigeons, for example, because they swallow seeds whole and need the grit to help hull the seeds). There seems to be a definite risk to overeating grit leading to impactions however, so this is something you should discyss with your avian veterinarian. If you do feed grit, only offer a few granules at a time and at infrequent intervals, especially mineralized grit.