Okay, we all want to help you. First of all, how long have you had your bird?
Birds take quite a while to adjust to new surroundings, new people, new voices, and in your case, also a new cage and lots of new toys.
That's alot for any bird to adjust to, and it will take you a while to gain your birds trust to have him step up onto you.
With a new bird, it's best just to sit near him without invading his "space" and scaring him. Talk quietly to him, read a book, sing songs, whatever you feel comfortable doing.
Key word here is Patience. You need to let your bird set the pace and get comfortable with all the "new" things.
Do not rush to train him yet.
Build trust first.Start offering treats through the cage bars.
As your wee one gets more comfortable, he will come out the cage on his own. You can put a perch on the outside of the cage and give him treats there.
With a healthy diet, a good routine, and time and patience you should literally have him eating out of your hands. It won't be today or tomorrow, and may take a few weeks to a few months, but with his wing feathers growing back, and his confidence growing, (as well as your own), you should have a great bond with him.
For your vet visit, a handkerchief or small towel would be good to transfer him to his carrier.
I personally would not ship an already stressed baby bird, but that's me. Just be patient, and read some training books or ask here. Someone will always offer advice for you.