Training during molting?

Given the fact you got your Lime 2 days ago, it's best to not start training so soon and to give him at least a couple of weeks for him to settle in his new home. For now, you can simply sit close to Lime's cage and work on establishing the foundations of trust by getting him used to your presence and voice, by talking to him in a calm, encouraging, positive and reassuring way.

As for any future training or close interaction with a moulting budgie, it's perfectly fine to keep up with it as long as the budgie is in good spirits and is actually receptive to having a closer interaction and genuinely wants to spend quality time with the owner.
During the moulting process there may be one or more days where the budgie prefers to have quiet time in the cage and this should be respected.
By watching the budgie's body language and behaviour, it should be easy to notice when they want to spend time with us or not and by making an interpretation we can then act according to it.

Congratulations on your little Lime!

Budgies are more stressed and crabby during moults, but that doesn't mean you have to lay off training completely. It's critical to watch his body language for signs that he's stressed or bothered, such as the feathers close to the body and the warning little squeaks that show annoyance.

First, however, he needs two weeks to settle in before you start taming him, so during that time he very well may get over his moult! During the time he's settling in, you should spend time just talking to him or sitting next to his cage so he can get used to you. If you build a foundation of trust early on, when you actually start putting your hand in the cage in two weeks, he will be more receptive :thumbsup:

EDIT: Aluz and I posted at the same time! I agree totally