だんだん is an adverb that is used to express the slight, but constant, progression of something. Originally, it comes from the kanji construction
段々（だんだん）, which can be thought of as meaning 'step by step', 'stepwise', or even 'steadily'.
だんだん is similar to
ほとんど, in that it can modify the meaning of entire phrases, rather than just single words.
To use
だんだん, simply put it before the phrase that you want to express as '(A) that is increasing steadily'.
12月（がつ）になってからだんだん寒（さむ）くなってきた。
Since December, it has gotten colder steadily.
だんだん仕事（しごと）の環境（かんきょう）に慣（な）れてきた。
I have steadily gotten used to work's environment.
While
だんだん can take the particle
と, it is not required. In fact,
と is very often omitted from adverbs/onomatopoeic words (words that represent a sound or motion) in Japanese.
Initially, it can be easy to mix-up
だんだん, with
どんどん.
どんどん is an onomatopoeic word that represents loud banging (like on a drum), so is used to express fast changes/progression. This is unlike
だんだん, which expresses slower changes/progression.
だんだん雨（あめ）が降（ふ）ってきた。
It is starting to rain. (Slow progression, does not imply great speed or severity)
どんどん雨（あめ）が降（ふ）ってきた。
It is starting to rain. (Rapid progression, may imply great speed and/or severity)