Tiny and delicate, the zebra finch isn't the most human-oriented of pet birds. That doesn't mean you can't train a zebra finch, but expect it to take time and considerable effort. If finger training is important to you, you might have the best results with a hand-raised bird. Trained or not, these attractive little birds are fun to watch.
Delay any type of training until your zebra finches are settled in their new cage. After a week or so, finches should acclimate to their surroundings and these high-strung little birds should relax somewhat. Start training by placing your hand into the cage several times daily, leaving it there for several minutes each time. You can try to offer the finch small treats, such as a bit of fruit. Eventually, a finch should come near your hand. At some point, it might alight on your finger.
Finger training takes time, but given a finch's life expectancy, he might share your home for up to 15 years. Patience is key. Once the finch comes to you, takes a treat and sits on your finger, keep repeating the process on a daily basis until it's a regular routine. Once he's reliably finger trained inside the cage, you can try taking training to the next step.
Out of the Cage
Once the finch reliably alights on your finger and appears comfortable sitting there, you can consider taking the bird out of the cage for brief periods. Before your first attempt and every time thereafter, finch-proof the room. That involves closing all doors and windows, removing any dogs or cats and turning off all fans. If the windows have blinds or curtains, close them. Keep the lights turned down low. Just keep the finch on your finger for a minute or two before putting him back in the cage. If he flies off, keep his cage door open, staying by it to prevent any other birds from escaping. He'll probably return soon to his friends and the safe haven offered by the cage.
If you decide to breed zebra finches, or have particularly good-looking specimens, you might get involved in bird exhibitions. Before taking your zebra finches to a show, they require some basic show training. Allow your finch pairs to inhabit a show cage -- smaller than a standard cage, with no toys and only a single perch -- for an hour at a time to get used to the setup. The Zebra Finch Society advises putting various birds together as pairs to see which ones appear most compatible. If one or both members of a pair start feather plucking, that's a sign that they're not exhibition-ready.