I had some trouble getting a white carnation in the beginning too, My tips are:
1: Place two pink carnations next to each other
2: Water them with a golden can
3: Burry fertilizer next to them
And repeat this process until you got your first one.
The second white one is exactly the same, only place a pink and a white carnation next to each other (that should grow another white one even quicker)
Now that you got your second white carnation, there's no other flower that can be bred when two white ones are standing next to each other.
Hope this helped, and yes, it might take a while before you get your first and second white carnation. But once you got them, it's pretty easy to make a whole field of them.