So it would be correct if you said "I am busy preparing to move." The article "a" can be used with a noun but not a verb. As for your second question, "Good luck with the move," is totally correct. You could also say, "Good luck with moving." In the second option you can't use "the" because it is an article. So, just like in the first example, you can't use an article with a verb.

I am busy preparing to move Or I am busy preparing for a move But the first one is more common I would think Good luck with the move is correct

For the good luck sentences both are correct but they have different meanings. Good luck with the move, means with the move from one house to another. Good luck with the moving, is saying good luck with the physical act if moving like packing boxes etc. At least that’s how I would interpret it.

You could say "I am busy preparing for a move" and "Good luck with the move" OR "Good luck moving." You could also say "I'm busy getting ready to move" and "Hope the move goes well!" Feel free to check out my teacher profile or send me a message if you're interested in booking a (paid) session with me sometime :) I provide detailed notes with corrections of mistakes made during the conversation, which my italki students find very helpful. Have a great day :)

I am busy preparing for a move. Good luck with the move.