I just got a parakeet a week ago from a pet store. When I first took him home, he let me pet him and didn't seem afraid. He even climbed on my finger for a little bit. Now, he will still eat millet from my hand but he hardly ever lets me pet him or put a perch under him. I don't know what I am doing wrong and I'm worried that I have ruined my relationship with him already.

leave him be for 10days an just stick to changing water an food by moving very slow when near or around him ...you should NOT be doing anything with him at this stage except letting him get use to his cage ...you...and surroundings

you never had a relationship with him in truth he was just way to confused with things to make up his own mind on what to do...

training takes a lot of time and a lot of patience and its all in bird time and you not forcing the bird or having any mirrors in the cage either.

nothing is ruined but can be if you try to make him do something he is not willing or ready to do at this time...eventually if you do things right he will perch on your hand then begin perch training very slowly and he will learn to do that ...its not accomplished in a couple days...

welcome to the forum

Many times new birds are too afraid NOT to put up a fuss and seem to willingly step on to your hand or finger. Then a few days later they realize, hey I can avoid them!
Like illusive said, you have not ruined anything yet. Just be patient, let him be without trying to get him to perch on your hand for a week or so. Just move slowly, stick to changing food/water and cleaning when necessary and talk to him lots and lots. It's alll new and scary to him and he needs time to get used to his new home and all that goes on there
Then begin putting your hand in cage, resting on a perch and just hold it still for 10 mins or so a few times a day. When he is comfortable with your hand just being in cage like that, then slowly move it a little closer to him talking all the while, and again stop and hold still. If he seems nervous and shies away stop where you are and be still for a while. Lesson over for that round and try again later.
It takes time and patience. Every bird is different so there really is no time frame. It's all in individual keet time.