Honeymoon in Europe (Dec-Jan) suggestions??

hi all!! My fiancé and I have booked our honeymoon trip to Europe and we will be there for a little bit over a month. We will be landing in London and from there.. who knows? I would like to go to Greece, but I am doubtful bc of the time of the year (in my mind Greece and winter dont go together but please tell me if i am wrong) do y’all have any recommendations on which cities we should definitely visit? cheap ways to get to different cities? where to spend Christmas? where to spend New Years? How cold is it?

We are looking for a fun trip more than a luxurious trip but splurging here and there is definitely in our plans.

Thanks!