Is my bird playing or stressed when it flips and drags in its cage?

He could just be bored or stretching. Giving him time outside of his cage to exercise, as well as providing plenty of enrichment such as toys, treats, time together, etc can help keep him in shape and reduce boredom.

He is bored. Provide a variety of toys and puzzle feeders for him. Increase play time, both in the cage and out, to keep him distracted. When the behavior is seen, a stern "No" should stop him and then he can be distracted. Be firm and be consistent.

Congratulations on your new birds! If the flying back and forth is in response to when you are putting your hand into the cage this could be due to fear or being startled. If he is randomly flying throughout the day with no relation to you approaching then it could be him just exercising. Millett is a treat that is very useful in helping gain your budgie's favor as well, so letting them nibble some from your hand or while you touch them if they are comfortable with you touching is helpful in making interaction with you a pleasant experience and can eventually be used to lure them onto your finger or hand!

It might be that John is just very excited/happy in that moment and is wanting to let everyone know. If he's otherwise acting healthy and normal I wouldn't be too worried. You can try offering a treat or placing a new toy in his cage to see if it helps keep him happy and interested in something to play with.

Without knowing the size of the cage, it is hard to give advice. If they have a large flight cage where they can easily move around and exercise, then that may be enough. Adding in some fun toys to explore, moving dishes and toys around or rotating them out, offering treats to chew on, etc can all help increase enrichment and make things more fun for your birds. If they're kept in very small cages, then looking for a playpen that can fit in the room or on top of the cage may be nice for your birds to climb on or play on. However, finches tend to explore more and may want to fly around, so it is best to make sure they are only let loose in rooms where they can easily be caught again and cannot fly into windows, doors, or other dangerous objects that can cause injury or death.

This sounds like your bird may be afraid of people. It can also be an indication of neurological disease. Without a physical exam, the cause cannot be determined. Consider an exam with your veterinarian or an experienced avian vet. If the bird is healthy, you can begin to desensitize it to your presence and begin hand taming. Start by simply sitting next tot he cage for short periods of time. Gradually increase the time as the bird settles down. Offer a small treat after each session. When the bird is calm, place your hand in the cage and start the cycle again. Short periods of time, increasing as the bird accepts you and end with a treat. Be patient and be consistent.

It sounds like your bird may not be comfortable yet with being handled around the cage. Stress and feeling unsafe can cause a bird to lash out, and it may be that they feel you are intruding on their space or are going to harm them when you get close. Taking some time to bond while your bird is in its cage may help. If he or she has a favorite treat, begin by offering it every time you approach the cage, and only at these times. This will help your bird start to associate your arrival with good things, rather than scary. Once used to you just tossing in a treat every time you come by, you can then move on to opening the cage and offering the treat from your hand, petting your bird, etc. If they become too scared, try to bite, or seem very stressed out, move back to the previous step that worked and try again from there. Making sure the cage itself is kept in a stress-free and quiet environment where other pets, people, etc can't bother it can also help reduce stress and make the space more comfortable and relaxing. Bonding does take time, and it may take days, weeks or even months on each step to build that trust! Be patient, and try not to be discouraged if you don't see immediate changes.