Improve my email below: 

Dear Professor,

I hope this email finds you well. I just wanted to wish you a very merry Christmas and a happy holiday season.

I also wanted to let you know that I have successfully started my PhD program and everything has been going well so far.

I was also wondering how you have been doing. I hope that you have been enjoying the holiday season and that everything is going well for you.

Best wishes,
Dear Professor [Last Name],

I hope you are doing well. As the holiday season approaches, I wanted to wish you a Merry Christmas and a joyful holiday.

I also wanted to share that I have successfully started my PhD program, and everything has been going smoothly so far.

It would be great to hear how you have been doing as well. I hope you are enjoying the holiday season and that all is progressing positively.

Best wishes,
[Your Full Name]