Hey! Sorry I know this is really late, but, I have some tips on how to train a finch because I have 2 zebra finches of my own.
So, first off, you want to know what type of personality your finch has. Observe how your finch reacts to you when you put your hand at the bottom of the cage. Once your figure out weather its curious (like my birds) or scared to death, thats where you want to start. If your finch is curious, start out by taking out the food for 30 to 1 hour. Then, cup your hand and put seeds in it, and stick it in the cage (where the food bowl was). The key is to wait, and be VERY patient. Soon your finch should start eating out of your hand! (mine even stepped up) Try to be very still, because sudden movements scare them. Now, if your finch is scared, hold you hand at the bottom of the cage for about 15 minutes a day. Soon, they shouldn't be as scared anymore. If they still won't eat from your hand, put millet spray (a treat you can find in local pet stores that your finch should love) at the bottom of the cage, but don't give them too much! Once you see your finch starts to eat it, after their done, try using that to let them eat from your hand. After they start eating from your hand, (do this for about a month) try to go near them without a treat. If it doesnt work, try and try again. Eventually, they wont care. Then day by day go closer and closer. But if they flip out, take a step back. Soon as your close enough, get the treats, distract them with the treats as they eat, then gently push on their belly, if they fly away,try again but no pushing. After 5 minutes, push again. Only do this 4 times a day. Finally, one day they'll step up from there and thats where all the training starts. Hope this helps! Good luck!
