Biggest Tip: Patience.
I’ve been lucky in that both of my babies were hand-raised so they’re already pretty used to being handled. But even with that you still have to build trust.
Spend time just near the cage, talking to them so they get used to your presence. Leave the cage door open so they can come out and investigate if they feel like it. LOTS of treats. I’m trying to regain Beckett’s trust right now after having to man-handle him to give him medication for almost a month. And it’s just a lot of offering him millet and letting him learn that hands=treats=good and that hands aren’t always going to be grabby.
Respect their boundaries. You are a big terrifying giant to them and they just need to learn that you’re part of their flock. If they run away from you, just stay still and see if they come back. If they’re nippy or aggressive, back off and try again later. To tie back to personal experience, sometimes when I ask Beckett to step up he just freezes, so I give it a second to see if he changes his mind, and then slowly pull my hand back.
There are hundreds of “Budgie Taming” guides online that you should definitely look into for more precise “guides” but the big things to keep in mind that will help things go smoothly for both of you is 1) you are Big™ and Scary™ and possibly a Predator™ and 2) it’s not “taming”, it’s trust building and bonding.
And always bring it back to Patience. If you get frustrated one day and back your bird up into a corner of the cage trying to grab them, that’s going to undo a lot of time spent building trust. (Now, sometimes you have to do that for their own good, vet visits and the like, but it should never just be “AH why won’t you come to me? *grab*)
Got a little rambley in there I think but I hope it helps a little.
Good luck!! If you haven’t already introduced the new bird, while she’s in quarantine is a great time to work on taming because you’ll have one-on-one time with her. And if she thinks you’re chill, it’ll make it easier to bring the other bird around.