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You have certainly come to the right place as you will find all you need to know on budgie care here.
When bringing home a new budgie, it's best to follow quarantine procedures by housing the new budgie in a separate cage and room away from your other budgie. This is done in order to prevent a potential illness carried by the new bird to be spread onto your other bird: https://talkbudgies.com/new-budgie-ar...necessary.html
As for the introductions, after quarantine time is over, it's advisable to place the newbie's cage close to your other budgie's cage in order for them both to get used to each other's presence. Once they show signs of curiosity and of wanting to meet, you can then allow your budgies to have closely supervised out of cage time together on neutral grounds (away from cages).
Depending on how well they get on, you can then decide to let them share the same cage together, they can even choose to do so on their own accord. Generally speaking, two male budgies will get along well and be very good friends. In the end it's all a matter of the budgie's personality. It's not a given that one male/female pair will get on well.
The detailed information on this link should be of use: https://talkbudgies.com/new-budgie-ar...o-budgies.html
Is your male budgie at the very least finger tamed? If so then you can make him step onto your finger/hand and gently lead him back to his cage.
For your new budgie, it will be trickier since chances are high she is not tame.
By moving very carefully as if you are in slow motion and placing her cage next to the Christmas tree, you could try to lure her back in.
You can place some millet by the cage's door, as well as a perch near it in order to facilitate your new budgie to get into the cage.
If this doesn't work, then you can use one of the perches and try to see if she steps up on it so that you can carefully lead her back inside the cage.
You can use the quarantine time to work on establishing a bond with your new budgie and to later on start the training process. For all the information on this matter, be sure to check the Taming and Bonding section of Talk Budgies.