Not sure that I remember what to expect. My little boy was already very friendly and seemed semi-tame when I got him yesterday, he learned step-up in about five minutes and happily wandered my room, occassionally going into his cage but willing to step out as soon as I opened the door. However, when night rolled around, I couldn't get him to sleep at all. His dropping were very runny. He does not seem to have slept since he's dozing off a bit right now, his droppings are still runny, and he hasn't touched his food or water. He's also suddenly scared of me, even though yesterday he was fine. I highly doubt he is sick, I am thinking it is still just him being nervous, but is there anything I should be wary of?

He is still getting adjusted to his new environment. The runny poop can be caused by stress. I'm sure in a few days his poops will become normal once he is settling in.

Do not worry about not seeing him drink water. Budgies only drink a small amount of water. He may be drinking or eating his seeds once you leave the room. If you are worried that he can't find his seeds, you can place a small bowl or plate and fill it with seeds at the bottom of his cage. Blow on the seeds and watch to see if any empty husks fly. If you do see any, means he is eating.

I would leave him in his cage and cover his cage on 3 sides. Leave the front uncovered. It will help him feel safe and secured in his cage.

At night do you cover his cage? Covering his cage will help him sleep.

I would wait at least a week until you start to work with taming him. You can still talk to him constantly and sit close by him. So he can get use to you. Here is a great article on how to tame your budgie Link. It is from the FAQ section.

When you first get a new budgie it is best to let them settle 3 or 4 days and let them get adjusted to their new enviroment. I would suggest sitting by the cage, talking to him but not handling as much just moving to your new home is stressful and though he may seem okay with it many birds will tolerate handling in the beginning because they are too fearful to react back and then in a week or so you may see more fearful behavior like bitting back, the dolphin noise etc... My suggestion is to keep him covered on all 3 sides for the next 4 days, let him get acclimated to his cage and you will see him come around and eat. Budgie drinking is different for every budgie. Merlin I never see drink Pretty I saw drink all the time. So as long as the poops are firm, white and black in the middle everything should be okay.