Best way to get my scared cockatiel use to hands?

I recently got my first bird. I bought a cockatiel from a local pet shop. I believe it is a male (normal grey. Yellow face with orange cheeks) I also think he is at least over a year old (he has no bars on any of his feathers). Though I'm not exactly sure how old he is.
I keep him in the bedroom as that is where I spend most of my time. I figured this would be the best way to get him use to me. I've had him for just shy of a month now, and he is still afraid of my hand. I feel bad because I haven't been able to take him to the vet to get him checked up and get his wings clipped so I can let him out of his cage more. I'm afraid that if I take him out with a towel or something to take him to the vet might make him more afraid.
He has gotten better though since I got him. Mainly what I do is place my hand in his cage and rest it on one of his perches. Most of the time he keeps his distance, but I have gotten him to walk over my hand a few times. If I get too close he acts like he will bite, but ultimately just moves away.
Is there anything else I can do to help him get more comfortable with my hand so he can have a better life and finally have some fun outside of his cage?

Hello, Congratulations to you and your new friend! Just shy of a month isn't all that long, but plenty of time for most tiels to adjust. If he's been in a pet store for a year he may be a little wild, but if he isn't thrashing about in his cage or shrieking in fear when you approach, then he is probably comfortable enough to interact more. Have you tried letting him come out of his cage in your bedroom? If your bedroom is parrot safe and you take some reasonable safety precautions, I don't see why you couldn't start letting him out now.

I've heard over and over again, watch your bird's body language. That's fine, but while you are watching your bird's body language, he or she is also watching yours. If you show signs of fear your bird will pick up on it immediately and use it against you. Sitting close to his cage and talking or reading to him is a good way to get to know your bird while he is busy studying you. A perch on the outside of the cage near the door will give your tiel some control and he will be less defensive. The first time you try this make sure your room is safe, cover the mirrors and windows, make sure no one will open the door while he is out. Stay as calm as possible, dim the lights, talk to him in a soft voice and let your parrot initiate contact. Make his out of cage time as enjoyable as possible.

There are many training methods to choose from, read up on all of them and choose the one that fits you and your bird the best. Be consistent and your bird will learn to trust you.

Thank you, I'll have to use one of his lower perches that he never uses and move it outside.
Also, speaking of body language. He likes to act tough. If I've got my hand in the cage or I'm trying to change his food and water. Whenever I take my hand out of the cage he'll come forward mouth open like he wants to bite, but if you put your hand in he just backs away.
I've just been ignoring him when he does this. I'm kinda worried my ignoring it might cause something negative? Am I handling it correctly?

Your handling it exactly right. Birds like drama if you make a big deal out of it then he will keep doing it. Some birds take forever to adjust some adjust right away so just keep giving him time. You can also use some millet and take some into your hand let him eat it out of your hand that way he starts seeing your hand as something that brings rewards rather than something to be afraid of.

Thanks everyone! Getting him out of his cage was a really good for progress. I started him out by getting him to step up onto a perch and after a while I moved to my hand. He even chilled out on my shoulder for a few hours! It's really great progress!

That's excellent progress! It sounds your new friend is doing fine. I love cockatiels, I think you'll find the tough guy routine is mostly bluff. He's probably thrilled to be allowed some freedom and companionship. Have you named him yet?

Yea, I think he's just pretending since he has bitten me a few times today while I was handling him and it wasn't all that hard. His name is Fabio

The only problem now is it's like all of progress gets reset in the morning

Sorry Fabio's so nippy but it's normal. He'll soon stop hitting the reset button, he just needs time to adjust to his new life. If the biting gets serious, you may need to back off and give him some space, then start fresh. It's part of the process.