How can I build trust with my budgie to safely take him out of his cage?

It may take time and patience. You can begin by simply sitting next to the cage for short periods of time initially and increasing it as they acclimate. They will flutter about and move to the far side of the cage at first. Eventually, they will settle down and accept your presence. Move slowly and speak softly around the cage. Once they have completely acclimated, you can progress to placing your hand in the cage. Make no attempt to touch or hold them. Again, they will flutter, panic and move to the far side of the cage. Leave your hand inside the cage for ever increasing periods of time. Use small treats as rewards after each session. Eventually, they will become curious and may hop onto your hand if you have treats. You can teach the 'step up' command by extending a finger and gently pressing it against the chest while saying "Perch" or "Step up". From this point, you can work with them to teach additional tricks or whistles.

Wait until they don't flutter about when you sit near the cage. Then place a hand I n the cage for ever increasing periods of time. They will fly away from you and screech when you place your hand in the cage. This is normal. Once they begin to settle down and accept your presence in the cage, place a small treat in your hand. Do this each time. They will eventually settle down and may approach your hand or finger for a treat. Be patient and be consistent. It may take weeks.

It may take a little longer than you thought to get him used to you. Some budgies, if they weren't hand tamed before you got them as little babies, they may always be a little skittish. Try putting his favorite snacks and foods on your hand and holding very still and see if comes over to nibble at that. Keep talking to him and singing to him softly all day so he gets used to you. Try sitting right next to him without moving as well.

Time and patience! It can take several months for a bird to start trusting enough to want to hop on you or interact more. It sounds like you're on the right track already with offering food from your hand. I would continue to do this, and gradually increase to petting Sky, and then handling her more/asking her to perch. If she becomes too nervous, letting her calm down and trying again later is best.

Time, patience, and lots of treats. If you notice Curio has a favorite food, you can start by offering that favorite food only when you're interacting with him. IE, when you pet him, approach his cage, let him out, etc. This can help him to associate you with something fun/good and also make him more curious every time you come by to spend time with him. The same can also be done with toys he likes to play with. Sitting near them when he's out of the cage playing, and feeding a treat every time he comes close or shows interest in you can help teach him that you are safe to interact with and play around. If at any time Curiou becomes nervous or scared, stopping the activity and letting him calm down is best. From there, you can go back to the previous way he was OK with (IE just giving a treat from your hand but not petting) and work again from there.

Time and patience are key to helping your parakeets get used to you, and each individual bird may bond differently to you due to their individual personalities. If your birds have a favorite treat, you can also use that by giving it to them every time you visit their cage or interact. I'm also including an in-depth article on teaching your bird to step up which can help with the bonding process as well: https://www.petcoach.co/article/basic-bird-training-how-to-train-your-bird-to-step-up/

Having a playpen or something your birds can rest on while out of the cage is a good idea. Also making sure they can get back into their cage for food and water (and to close it when you're ready for them to be back in) is also best. You may also want to have a bird net on hand to catch them if they aren't super tame and won't go back into their cage. Clipping wings can help make them easier to handle and catch, and if you are unsure how to do this, having a local vet show you before allowing them out and about is best. Making sure the room doesn't have any harmful objects such as fans, open windows, cooking items, etc is best to avoid injury to your birds. If there are sharp objects or things they could be injured on, removing them before allowing your birds out is best.