Congrats on the new birdie! What kind of parakeet did you get?
As far as the warming up thing - it does take time for them to adjust. How old is the bird? Several 20 minute sessions a day would be great. He just needs love and attention to come around. If it's a rescue or a re-home then it will take more work then if you have a hand raised baby. It sounds like you're doing great though, if he will sit on your hand for short periods of time. Just continue to do what you are doing. Be sure to talk to him...be animated and expressive (even if you feel silly!) and feel free to offer him treats!
The noise is normal - that's a bird for ya!
Most, but not all, species of birds are more vocal or talkative or "noisy" during the morning and evening. And he will eventually discover his cuttlebone. In one of your times when he is sitting on your finger you can show it to him....you can also show him some of his toys. Be sure to talk goofy - alot of times the tones of our voices can get their attention.
Is this bird a baby? Kinda seems like it might be. If it's just a baby, give it time, it will come around.
Here is a link on wing trimming: Wing Trimming
(it's for lovebirds but you get the idea)
Here is a link on nail trimming:
I would strongly suggest taking your bird to get groomed a few times (vet's do it for pretty cheap as do bird stores) and watching how they do it. You can ask them questions, etc. It can be scary the first time you do it and this way you'll feel more comfortable.
Good luck with your new birdie!