Okay, first of all DO NOT CHASE HIM AROUND THE ROOM OR USE A TOWEL OR "CLOTHES BASKET"! You need to ear his trust before anything else, and every time you chase him or throw a towel, clothes basket, etc. over him you're scaring him more and more. So the way to avoid this, to make him rely upon you to help him, and to make him listen to you and stay in close quarters or near you without trying to get away from you is by getting his wings clipped ASAP!
I've tamed a ton of Petco and PetSmart parakeets. I bred my own English and American budgies for 20 years and I had multiple friends, family, and their friends bring me the wild parakeets they bought elsewhere
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So the very first thing I do is clip the outermost 4-5 primary flight feathers only on each wing. This will keep them from flying but still allow them to slowly glide to the floor in a controlled manner. What does this do for you? First of all and most importantly it forces the bird to stay near you when you're trying to work on hand taming it or when you're simply trying to spend time with it outside of the cage. It's pretty impossible to tame a bird or to get a bird to start feeling comfortable around you when it's constantly flying away from you the second you come within 10 feet of it. After you clip him and you allow him out of his cage, you'll be able to sit near him and talk to him, read to him, etc. And you will never have to chase him around the room again or throw something over him to catch him again. Instead, he will be absolutely forced to rely upon you to get back to his cage! So let's say you let him out of his cage and he immediately tries to fly away the first time, but instead he'll glide to the floor. You can sit and talk to him while he sits on the floor, while he explores things, etc. When he's ready to go back to his cage you'll be amazed how easily you get him to step up onto a wooden perch (which is a really good first hand-taming step). Just take a long perch, long yard stick, etc. and very slowly lower it to above his feet, gently touching his lower belly. It may take a few tries, in fact this is a great first taming lesson for you to do with him, eventually he'll step-up onto the perch and you can slowly move him back to his cage. After you accomplish this a couple of times he'll realize that he needs your help. This very quickly will lead to you using your finger to have him step-up onto instead of the perch. Some people go straight to trying their finger, but rushing him won't earn trust. Go at his pace, they are all different.
The millet trick for hand-taming is another great training exercise, it's usually done while your bird is inside his cage. However, with his wings clipped you can practice outside the cage as well. Before getting to the millet, the first step (which is boring, but necessary) is to put a chair next to his cage, sit down, open the door, place your hand inside his cage but not near him, just right inside, and just keep it there without moving it for a good 20-30 minutes while at the same time continually talking to him or reading a book to him in a soft, gentle voice. That's it. The key is getting him used to your voice and associating it with your hand, and learning that your hand is safe. I usually do this as many times a day as possible, at least 3-4. The idea is that after a few days of him used to the distance your hand is at, move it slowly closer to him until he shows he's scared, then stop and keep your hand there at that distance for a few days, and then get closer, etc. When you get your hand close enough to him that he can reach a millet spray, go ahead and hold one in your hand while you are doing the training exercise. After he is willing to eat off of the millet spray, then each time you do it you need to stick out less and less of the millet spray, making him get closer and closer to your hand in order to eat it, and eventually he'll have to step into your hand to eat it. Once he's standing on your hand willingly, you can actually place a millet spray in your hand and have your index finger sticking out with the millet spray only accessible to him if he steps onto your finger. So you place your index finger slowly and gently above his feet, touching his lower belly. He'll do a lot of stepping on and right back off, both on the perch and your hand. That's okay, let him, just keep the perch or your hand steady, don't move, and eventually he'll step-up.
The main idea here is in order to hand-tame a wild bird you need to earn his trust, and the only way to do that is to spend as much time with him one-on-one as you can, as often as you can, getting him used to being near you, used to hearing the sound of your voice often, and used to your hands being near him but not scaring him or hurting him. This is why the chasing him around the room and especially throwing things on top of him and grabbing him to get him back in his cage is very, very counterproductive and sets you back in a negative direction. And this is why you need to clip his outermost 4-5 primary flight feathers only.
I have only ever had to clip my bird's wings one time when I first brought them home. I would clip them the minute I got them home and start training right away. Their feathers will fully grow back in about 3 months or so, and by then, if you commit to working with him every single day for as much time as possible, spending as much time with him as you possibly can, you will not need to clip his wings again. He will be able to fully fly around your house, stay on top of his cage or on a playstand, or on your shoulder while you're in the room with him watching TV, reading, on the computer, etc. And he'll fly to you when you call him and go back to his cage on his own for you. He'll love hanging out with you and eventually bond with you. That's the goal. But you'll never accomplish this unless you're able to gain control over him immediately, right off the bat.
If you don't know how to properly clip his wings you have some options. Any avian vet can do it, most regular vets can do it. Any pet shop that sells birds should be able to do it, though you must specify that they only clip the outermost primary flight feathers only, otherwise you can end up with a really bad clip that makes your bird drop to the ground like a rock. This can severely injure him, plus it will then take forever for them to grow back. I do my own wing clipping, I have since I was like 8 years old, it's very easy once you learn. There are many videos showing you how to do it step-by-step on YouTube, watch them and see if you feel comfortable doing it, and if you have any questions just ask. The only thing you must check for before you clip his wings is if any of his primary feathers that you'll be clipping are BLOOD FEATHERS. The YouTube videos go over this and there are a ton of online tutorials with photos. And yes, you'll have to wrap him up in a towel in order to clip his wings this time, or if a vet or groomer does it. That's OK, it's necessary this time. And as long as you aren't chasing him first and you get him in the towel right out of his cage, he'll be mad at the towel, not you.
"Dance like nobody's watching..."
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