Hi Lily and
to the forums!
Congratulations on your new budgie! I'm glad you've been doing all the necessary research to give him the best life possible
You've come to the very best place to continue your research!
It's normal for him not to eat or drink in front of you for the first days of him being home. If there are droppings at the bottom of the cage, he has been eating. You probably won't be able to tell if he's drinking since budgies drink so little, but he is. The beak grinding is a good sign, it means he's settling in well!
Singing, reading, and talking to him as much as possible during the next few weeks will help him feel comfortable with you and your family. Remember not to put your hand in the cage for at least two weeks as he needs time to get used to his cage and environment without any other stressors.
I'm glad you've recognized that he needs a larger cage, and transferring him shouldn't be too difficult. There's no reason to touch him to transfer him to a new cage, you can transfer his food and water dish along with several perches to the new cage and face the two cages so that they're touching, with both entrances lined up. You may need to raise/lower each cage by putting it on a stack of sturdy books or a tall table so that they are lined up properly. If there's a small space between the cages, put a sheet or towel over the part where the cages line up to discourage him from trying to squeeze through. Eventually, he'll get hungry or thirsty and will make it over to the new cage.
Despite precautions, taking care that the room is budgie safe, all other pets are out of the room, and all doors and windows are closed will ensure that if he does escape during the process, stress is minimized for everyone involved until you're able to coax him onto a perch or gently pick him up in a soft cloth and transfer him to his new cage.
I hope this helps!
If you have any questions after reading through the forums' many articles and stickies, be sure to ask!
We look forward to meeting Donny soon!