どんどん (or ドンドン) is one of the countless words in Japanese that represent a particular sound or sensation. These words are known as onomatopoeia (オノマトペ in Japanese).
どんどん represents the sound of banging (like on a drum), or the pounding of feet. In this way, it is regularly translated as ‘rapidly’, or ‘quickly’.
Like
だんだん,
どんどん may be used before almost any phrase.
-
彼女（かのじょ）が出来（でき）てからどんどんお金（かね）が減（へ）っていく。
Since I got a girlfriend, my cash has been rapidly decreasing.
-
どんどん頼（たの）んでね！
Quickly order! (Literally means ‘more and more’ in a sentence like this, with the nuance of ‘don’t hold back, order in rapid succession’)
While
どんどん can take the particle
と, it is not required. In fact,
と is very often omitted from adverbs/onomatopoeic words in Japanese.
Initially, it can be easy to mix-up
どんどん, with
だんだん.
だんだん is used to express slower changes, and may be thought of as similar to ‘steadily’, or ‘step by step’ in English.
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毎日（まいにち）ピアノを練習（れんしゅう）しているから、だんだんピアノが上手（じょうず）になってきた。
Because I practice the piano everyday, I have steadily gotten better at playing. (Slow progression)
-
毎日（まいにち）ピアノを練習（れんしゅう）しているから、どんどんピアノが上手（じょうず）になってきた。
Because I practice the piano everyday, I have rapidly gotten better at playing. (Rapid progression)
When the と particle is used with onomatopoeic words, it is very similar to one of と’s main usages as a quotation marker in Japanese. This means that と has a similar role to words like ‘went’, or ‘goes’ in English. For example, In English, we would say:
The door ‘goes’ BANG with the wind.
However, the door doesn’t actually go anywhere. This is similar to what と is doing here.