(Not a Teacher)
In the case of your example, it's:
"After weeks of preparation and training, they started to climb."
But if "weeks of preparation and training" were the subject, then a "the" would be required:
"The weeks of training and preparation were crucial to the success of the climb."
However, as I am not an English teacher, I don't really know the hard and fast rules on "the", assuming there are any.
Student or Learner