Welcome to the forums and the ever exciting world of budgie ownership!
You have certainly come to the right place as you will find all you need to know on budgie care here.
It's good that you and your family have been researching on how to take good care of Moonbeam and surely the information on the several articles and sticky threads at the top of each section of the forums will be useful and of benefit to you and your new little friend.
If you post more of a close up picture, a front shot of the whole beak/cere area in natural lighting (no flash from camera), we can help in identifying Moonbeam's gender.
From the little I'm seeing on the photo you posted, the cere seems to be mainly whitish in colour, if that is the case, then you have a young female.
As for the training and while it's great that Moonbeam has managed to relax a bit after the first couple of days of being home with you, it's still too soon to take a more hands on approach in training.
Generally speaking, it's best to wait at least a couple of weeks, allow the new budgie to fully settle into the house and allow him/her the time to get to know their family too, of getting the budgie used to your presence/voice and allow the bird to realise you are not a threat.
This is how the foundations of trust are built, by spending time close to the budgie's cage, talking to him/her in a calm, positive, encouraging and reassuring way. With time, the pet bird will give clues for you to take the next step in the actual training and this can be seen once the budgie shows signs of receptiveness towards you, moves closer to you while in the cage and you are able to notice a connection with the budgie.
This is when you can slowly introduce your hand and at first try to offer a little treat to the cage's bars, this will make your budgie associate the hand with something good. Later on, you can start to place your hand inside the cage with the intent of teaching the step up command.
In order to avoid things from getting too overwhelming, it would be better to take the training process more slowly.
For all the detailed information, you can check the stickies at the top of the Taming and Bonding section's main page.
Best of luck!