Hello and welcome Robin, Chris and CJ! As the others have said, you need to slow down and lower your expectations. It takes a long time (think months or years) for a rehome/rescue to trust you. That first week is what we call the "honeymoon" period. Then it seems like everything is moving backwards for awhile, just like you noticed. That is very common and is what happened to me. After the honeymoon week, for the first 2-3 months, I thought I had made a horrible mistake and Timber was doomed to be a "cage bird" with no interaction with us. I kept trying the suggestions given here, like speaking sofly to him, telling him what I was doing, whistling to him, opening his cage door and letting him come out on his own, sitting quietly by his cage and talking to him. touching his beak gently through the cage bars, and slowly he began coming around. It has been almost a year now, and we have developed a pretty good relationship. He will step up if he is out of the cage, and he looks forward to some head scratching at the end of the day as he sits on my arm. If you had told me we we get to this point in those first three months I had him, I wouldn't have believed it.
Hopefully someone with more experience will give you suggestions for getting him to the vet. If he seems healthy, you might want to give him more time to adjust before you stuff him in a carrier and take him somewhere he doesn't want to go. If you suspect any illness, that's another story. Timber still stays "mad" for a couple of weeks after a vet visit for nail trimming.